On primary Carmichael numbers

TL;DR

This paper studies primary Carmichael numbers, a special subset of Carmichael numbers with unique digit sum properties, exploring their construction, properties, and connections to taxicab and polygonal numbers, including Ramanujan's 1729.

Contribution

It introduces the concept of primary Carmichael numbers, analyzes their structure via polynomial constructions, and establishes their properties and connections to classical number theory objects.

Findings

All Carmichael numbers with three factors can be generated by specific polynomials.

Primary Carmichael numbers satisfy digit sum conditions for all prime factors.

Connections between primary Carmichael numbers and taxicab/polygonal numbers are established.

Abstract

The primary Carmichael numbers were recently introduced as a special subset of the Carmichael numbers. A primary Carmichael number $m$ has the unique property that $s_p(m) = p$ holds for each prime factor $p$, where $s_p(m)$ is the sum of the base-$p$ digits of $m$. The first such number is Ramanujan's famous taxicab number $1729$. Due to Chernick, all Carmichael numbers with three factors can be constructed by certain squarefree polynomials $U_3(t) \in \mathbb{Z}[t]$, the simplest one being $U_3(t) = (6t+1)(12t+1)(18t+1)$. We show that the values of any $U_3(t)$ obey a special decomposition for all $t \geq 2$ and besides certain exceptions also in the case $t=1$. These cases further imply that if all three factors of $U_3(t)$ are simultaneously odd primes, then $U_3(t)$ is not only a Carmichael number, but also a primary Carmichael number. Together with the exceptional cases, all Carmichael numbers with three factors have at least the property that $s_p(m) = p$ holds for the greatest prime factor $p$ of $m$. Subsequently, we show some connections to taxicab and polygonal numbers, involving the number $1729$ as an example again.

Tables

Table 1. Table 1.1. Distributions of C ​ ( x ) 𝐶 𝑥 C(x) , C ′ ​ ( x ) superscript 𝐶 ′ 𝑥 C^{\prime}(x) , and their subsets. The ratios are rounded to three decimal places.

$x$	$C\left(x\right)$	$C_3\left(x\right)$	$C^{\prime }\left(x\right)$	$C_3^{\prime }\left(x\right)$	$C_4^{\prime }\left(x\right)$	$C_5^{\prime }\left(x\right)$	$C_3^{\prime }/C^{\prime }\left(x\right)$	$C_3^{\prime }/C_3\left(x\right)$
${10}^3$	$1$	$1$					—	—
${10}^4$	$7$	$7$	$2$	$2$			$1.000$	$0.286$
${10}^5$	$16$	$12$	$4$	$4$			$1.000$	$0.333$
${10}^6$	$43$	$23$	$9$	$9$			$1.000$	$0.391$
${10}^7$	$105$	$47$	$19$	$19$			$1.000$	$0.404$
${10}^8$	$255$	$84$	$51$	$48$	$3$		$0.941$	$0.571$
${10}^9$	$646$	$172$	$107$	$104$	$3$		$0.972$	$0.605$
${10}^{10}$	$1547$	$335$	$219$	$214$	$5$		$0.977$	$0.639$
${10}^{11}$	$3605$	$590$	$417$	$409$	$8$		$0.981$	$0.693$
${10}^{12}$	$8241$	$1000$	$757$	$741$	$16$		$0.979$	$0.741$
${10}^{13}$	$\mathrm{19\hspace{0.17em}279}$	$1858$	$1470$	$1433$	$37$		$0.975$	$0.771$
${10}^{14}$	$\mathrm{44\hspace{0.17em}706}$	$3284$	$2666$	$2599$	$67$		$0.975$	$0.791$
${10}^{15}$	$\mathrm{105\hspace{0.17em}212}$	$6083$	$5040$	$4896$	$144$		$0.971$	$0.805$
${10}^{16}$	$\mathrm{246\hspace{0.17em}683}$	$\mathrm{10\hspace{0.17em}816}$	$9280$	$8996$	$282$	$2$	$0.969$	$0.832$
${10}^{17}$	$\mathrm{585\hspace{0.17em}355}$	$\mathrm{19\hspace{0.17em}539}$	$\mathrm{17\hspace{0.17em}210}$	$\mathrm{16\hspace{0.17em}694}$	$514$	$2$	$0.970$	$0.854$
${10}^{18}$	$\mathrm{1\hspace{0.17em}401\hspace{0.17em}644}$	$\mathrm{35\hspace{0.17em}586}$	$\mathrm{32\hspace{0.17em}039}$	$\mathrm{31\hspace{0.17em}103}$	$933$	$3$	$0.971$	$0.874$

Table 2. Table 2.1. Distributions of C ′ ​ ( x ) superscript 𝐶 ′ 𝑥 C^{\prime}(x) , C ​ ( x ) 𝐶 𝑥 C(x) , S ∗ ′ ​ ( x ) superscript subscript 𝑆 ′ 𝑥 S_{*}^{\prime}(x) , S ∗ ​ ( x ) subscript 𝑆 𝑥 S_{*}(x) , S ′ ​ ( x ) superscript 𝑆 ′ 𝑥 S^{\prime}(x) , S ​ ( x ) 𝑆 𝑥 S(x) , S ′ ¯ ​ ( x ) ¯ superscript 𝑆 ′ 𝑥 \overline{S^{\prime}}(x) , and S ¯ ​ ( x ) ¯ 𝑆 𝑥 \overline{S}(x) .

$x$	$C^{\prime }\left(x\right)$	$C\left(x\right)$	$S_{\ast }^{\prime }\left(x\right)$	$S_{\ast }\left(x\right)$	$S^{\prime }\left(x\right)$	$S\left(x\right)$	$\overline{S^{\prime }}\left(x\right)$	$\overline{S}\left(x\right)$
${10}^1$							$1$	$1$
${10}^2$			$1$	$5$	$2$	$5$	$32$	$60$
${10}^3$		$1$	$5$	$53$	$9$	$56$	$220$	$742$
${10}^4$	$2$	$7$	$13$	$477$	$34$	$532$	$1401$	$8050$
${10}^5$	$4$	$16$	$32$	$4147$	$100$	$4837$	$8388$	$\mathrm{84\hspace{0.17em}057}$
${10}^6$	$9$	$43$	$62$	$\mathrm{35\hspace{0.17em}827}$	$254$	43 981	$\mathrm{51\hspace{0.17em}333}$	$\mathrm{864\hspace{0.17em}438}$

Table 3. Table 3.1. Distributions of C ♯ ​ ( x ) superscript 𝐶 ♯ 𝑥 C^{\sharp}(x) and C 4 ♯ ​ ( x ) , … , C 11 ♯ ​ ( x ) subscript superscript 𝐶 ♯ 4 𝑥 … subscript superscript 𝐶 ♯ 11 𝑥 C^{\sharp}_{4}(x),\dots,C^{\sharp}_{11}(x) .

$x$	$C^{♯}\left(x\right)$	$C_4^{♯}\left(x\right)$	$C_5^{♯}\left(x\right)$			$\dots$			$C_{11}^{♯}\left(x\right)$
${10}^9$	$11$	$1$	$7$	$3$
${10}^{10}$	$48$	$1$	$19$	$27$	$1$
${10}^{11}$	$169$	$1$	$49$	$94$	$25$
${10}^{12}$	$590$	$1$	$104$	$346$	$135$	$4$
${10}^{13}$	$1780$	$1$	$194$	$899$	$622$	$63$	$1$
${10}^{14}$	$5456$	$2$	$397$	$2326$	$2252$	$456$	$23$
${10}^{15}$	$\mathrm{16\hspace{0.17em}245}$	$2$	$692$	$5482$	$7504$	$2420$	$145$
${10}^{16}$	$\mathrm{47\hspace{0.17em}171}$	$3$	$1227$	$\mathrm{12\hspace{0.17em}149}$	$\mathrm{22\hspace{0.17em}287}$	$\mathrm{10\hspace{0.17em}293}$	$1189$	$23$
${10}^{17}$	$\mathrm{136\hspace{0.17em}704}$	$3$	$2205$	$\mathrm{26\hspace{0.17em}464}$	$\mathrm{61\hspace{0.17em}640}$	$\mathrm{38\hspace{0.17em}886}$	$7187$	$318$	$1$
${10}^{18}$	$\mathrm{386\hspace{0.17em}066}$	$4$	$3713$	$\mathrm{54\hspace{0.17em}128}$	$\mathrm{158\hspace{0.17em}276}$	$\mathrm{131\hspace{0.17em}641}$	$\mathrm{35\hspace{0.17em}472}$	$2785$	$47$

Table 4. Table 3.2. Distributions of C ​ ( x ) 𝐶 𝑥 C(x) and C 3 ​ ( x ) , … , C 11 ​ ( x ) subscript 𝐶 3 𝑥 … subscript 𝐶 11 𝑥 C_{3}(x),\dots,C_{11}(x) .

$x$	$C\left(x\right)$	$C_3\left(x\right)$	$C_4\left(x\right)$	$C_5\left(x\right)$	$\dots$					$C_{11}\left(x\right)$
${10}^3$	$1$	$1$
${10}^4$	$7$	$7$
${10}^5$	$16$	$12$	$4$
${10}^6$	$43$	$23$	$19$	$1$
${10}^7$	$105$	$47$	$55$	$3$
${10}^8$	$255$	$84$	$144$	$27$
${10}^9$	$646$	$172$	$314$	$146$	$14$
${10}^{10}$	$1547$	$335$	$619$	$492$	$99$	$2$
${10}^{11}$	$3605$	$590$	$1179$	$1336$	$459$	$41$
${10}^{12}$	$8241$	$1000$	$2102$	$3156$	$1714$	$262$	$7$
${10}^{13}$	$\mathrm{19\hspace{0.17em}279}$	$1858$	$3639$	$7082$	$5270$	$1340$	$89$	$1$
${10}^{14}$	$\mathrm{44\hspace{0.17em}706}$	$3284$	$6042$	$\mathrm{14\hspace{0.17em}938}$	$\mathrm{14\hspace{0.17em}401}$	$5359$	$655$	$27$
${10}^{15}$	$\mathrm{105\hspace{0.17em}212}$	$6083$	$9938$	$\mathrm{29\hspace{0.17em}282}$	$\mathrm{36\hspace{0.17em}907}$	$\mathrm{19\hspace{0.17em}210}$	$3622$	$170$
${10}^{16}$	$\mathrm{246\hspace{0.17em}683}$	$\mathrm{10\hspace{0.17em}816}$	$\mathrm{16\hspace{0.17em}202}$	$\mathrm{55\hspace{0.17em}012}$	$\mathrm{86\hspace{0.17em}696}$	$\mathrm{60\hspace{0.17em}150}$	$\mathrm{16\hspace{0.17em}348}$	$1436$	$23$
${10}^{17}$	$\mathrm{585\hspace{0.17em}355}$	$\mathrm{19\hspace{0.17em}539}$	$\mathrm{25\hspace{0.17em}758}$	$\mathrm{100\hspace{0.17em}707}$	$\mathrm{194\hspace{0.17em}306}$	$\mathrm{172\hspace{0.17em}234}$	$\mathrm{63\hspace{0.17em}635}$	$8835$	$340$	$1$
${10}^{18}$	$\mathrm{1\hspace{0.17em}401\hspace{0.17em}644}$	$\mathrm{35\hspace{0.17em}586}$	$\mathrm{40\hspace{0.17em}685}$	$\mathrm{178\hspace{0.17em}063}$	$\mathrm{414\hspace{0.17em}660}$	$\mathrm{460\hspace{0.17em}553}$	$\mathrm{223\hspace{0.17em}997}$	$\mathrm{44\hspace{0.17em}993}$	$3058$	$49$

Table 5. Table 4.1. Examples of U 𝐫 ​ ( 0 ) subscript 𝑈 𝐫 0 U_{\mathbf{r}}(0) and U 𝐫 ​ ( 1 ) subscript 𝑈 𝐫 1 U_{\mathbf{r}}(1) .

$𝐫$	$(\tau ,t)$	$\vartheta$	value		decomposition
$(1,2,3)$	$(1,0)$	$2$	$U_{𝐫}\left(0\right)$ $=$	$1$	$1\cdot 1\cdot 1$	$\notin 𝔖$
$(1,2,7)$	$(2,0)$	$2$	$U_{𝐫}\left(0\right)$ $=$	$225$	$3\cdot 5\cdot 15$	$\notin 𝔖$
$(2,7,13)$	$(1,0)$	$6$	$U_{𝐫}\left(0\right)$ $=$	$\mathrm{13\hspace{0.17em}833}$	$9\cdot 29\cdot 53$	$\in 𝔖\setminus {𝔖}^{\prime }$
$(1,2,7)$	$(2,1)$	$2$	$U_{𝐫}\left(1\right)$ $=$	$\mathrm{63\hspace{0.17em}393}$	$17\cdot 33\cdot 113$	$\in 𝔖\setminus {𝔖}^{\prime }$
$(1,3,5)$	$(1,0)$	$9$	$U_{𝐫}\left(0\right)$ $=$	$\mathrm{29\hspace{0.17em}341}$	$13\cdot 37\cdot 61$	$\in {𝔖}^{\prime }\cap {𝒞}_3^{\prime }$
$(2,3,5)$	$(1,0)$	$26$	$U_{𝐫}\left(0\right)$ $=$	$\mathrm{252\hspace{0.17em}601}$	$41\cdot 61\cdot 101$	$\in {𝔖}^{\prime }\cap {𝒞}_3^{\prime }$

Table 6. Table 4.2. Chernick’s examples of U 𝐫 ​ ( t ) subscript 𝑈 𝐫 𝑡 U_{\mathbf{r}}(t) .

$𝐫$	$U_{𝐫}\left(t\right)$	$({\sigma }_1,{\sigma }_2,{\sigma }_3,\ell ,\tau )$
$(1,2,3)$	$\left(6t+1\right)\left(12t+1\right)\left(18t+1\right)$	$(6,11,6,0,1)$
$(1,2,5)$	$\left(10t+7\right)\left(20t+13\right)\left(50t+31\right)$	$(8,17,10,6,1)$
$(1,3,8)$	$\left(24t+13\right)\left(72t+37\right)\left(192t+97\right)$	$(12,35,24,12,1)$
$(2,3,5)$	$\left(60t+41\right)\left(90t+61\right)\left(150t+101\right)$	$(10,31,30,20,1)$

Table 7. Table 4.3. Further examples of U 𝐫 ​ ( t ) subscript 𝑈 𝐫 𝑡 U_{\mathbf{r}}(t) .

$𝐫$	$U_{𝐫}\left(t\right)$	$({\sigma }_1,{\sigma }_2,{\sigma }_3,\ell ,\tau )$
$(1,2,7)$	$\left(14t+3\right)\left(28t+5\right)\left(98t+15\right)$	$(10,23,14,2,2)$
$(1,3,4)$	$\left(12t+5\right)\left(36t+13\right)\left(48t+17\right)$	$(8,19,12,4,1)$
$(1,3,5)$	$\left(15t+13\right)\left(45t+37\right)\left(75t+61\right)$	$(9,23,15,12,1)$
$(2,7,13)$	$\left(364t+9\right)\left(1274t+29\right)\left(2366t+53\right)$	$(22,131,182,4,1)$

Table 8. Table 4.4. Example of U 𝐫 ​ ( 1 ) ∈ 𝒞 3 ′ subscript 𝑈 𝐫 1 subscript superscript 𝒞 ′ 3 U_{\mathbf{r}}(1)\in\mathcal{C}^{\prime}_{3} .

$𝐫$	$(101,199,499)$
	$\left(\mathrm{1\hspace{0.17em}012\hspace{0.17em}969\hspace{0.17em}501}t+\mathrm{557\hspace{0.17em}673\hspace{0.17em}420}\right)$
$U_{𝐫}\left(t\right)$	$\times \left(\mathrm{1\hspace{0.17em}995\hspace{0.17em}850\hspace{0.17em}799}t+\mathrm{1\hspace{0.17em}098\hspace{0.17em}782\hspace{0.17em}282}\right)$
	$\times \left(\mathrm{5\hspace{0.17em}004\hspace{0.17em}671\hspace{0.17em}099}t+\mathrm{2\hspace{0.17em}755\hspace{0.17em}237\hspace{0.17em}982}\right)$
$({\sigma }_1,{\sigma }_2,{\sigma }_3,\ell ,\tau )$	$(799,\mathrm{169\hspace{0.17em}799},\mathrm{10\hspace{0.17em}029\hspace{0.17em}401},\mathrm{5\hspace{0.17em}521\hspace{0.17em}519},1)$

Table 9. Table 5.1. First numbers m = U 𝐫 ​ ( 0 ) ∈ 𝒞 3 ′ 𝑚 subscript 𝑈 𝐫 0 subscript superscript 𝒞 ′ 3 m=U_{\mathbf{r}}(0)\in\mathcal{C}^{\prime}_{3} .

$m$	$𝐫$	$(\tau ,t)$		$m$	$𝐫$	$(\tau ,t)$
$2821$	$(1,2,5)$	$(1,0)$		$\mathrm{14\hspace{0.17em}469\hspace{0.17em}841}$	$(4,21,29)$	$(1,0)$
$\mathrm{29\hspace{0.17em}341}$	$(1,3,5)$	$(1,0)$		$\mathrm{15\hspace{0.17em}247\hspace{0.17em}621}$	$(1,3,23)$	$(1,0)$
$\mathrm{46\hspace{0.17em}657}$	$(1,3,8)$	$(1,0)$		$\mathrm{15\hspace{0.17em}829\hspace{0.17em}633}$	$(1,13,16)$	$(2,0)$
$\mathrm{252\hspace{0.17em}601}$	$(2,3,5)$	$(1,0)$		$\mathrm{17\hspace{0.17em}236\hspace{0.17em}801}$	$(5,7,18)$	$(1,0)$
$\mathrm{1\hspace{0.17em}193\hspace{0.17em}221}$	$(1,2,21)$	$(1,0)$		$\mathrm{17\hspace{0.17em}316\hspace{0.17em}001}$	$(1,3,40)$	$(2,0)$
$\mathrm{1\hspace{0.17em}857\hspace{0.17em}241}$	$(1,6,11)$	$(2,0)$		$\mathrm{29\hspace{0.17em}111\hspace{0.17em}881}$	$(3,4,7)$	$(1,0)$
$\mathrm{5\hspace{0.17em}968\hspace{0.17em}873}$	$(1,3,26)$	$(2,0)$		$\mathrm{31\hspace{0.17em}405\hspace{0.17em}501}$	$(1,9,10)$	$(1,0)$
$\mathrm{6\hspace{0.17em}868\hspace{0.17em}261}$	$(1,5,18)$	$(2,0)$		$\mathrm{34\hspace{0.17em}657\hspace{0.17em}141}$	$(19,42,43)$	$(1,0)$
$\mathrm{7\hspace{0.17em}519\hspace{0.17em}441}$	$(1,6,19)$	$(2,0)$		$\mathrm{35\hspace{0.17em}703\hspace{0.17em}361}$	$(5,23,176)$	$(1,0)$
$\mathrm{10\hspace{0.17em}024\hspace{0.17em}561}$	$(7,27,52)$	$(1,0)$		$\mathrm{37\hspace{0.17em}964\hspace{0.17em}809}$	$(2,7,17)$	$(1,0)$

Table 10. Table 5.2. First numbers m = U 𝐫 ​ ( t ) ∈ 𝒞 3 ∖ 𝒞 3 ′ 𝑚 subscript 𝑈 𝐫 𝑡 subscript 𝒞 3 subscript superscript 𝒞 ′ 3 m=U_{\mathbf{r}}(t)\in\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3} .

$m$	$𝐫$	$(\tau ,t)$		$m$	$𝐫$	$(\tau ,t)$
$561$	$(1,5,8)$	$(2,0)$		$\mathrm{314\hspace{0.17em}821}$	$(1,5,33)$	$(2,0)$
$1105$	$(1,3,4)$	$(1,0)$		$\mathrm{334\hspace{0.17em}153}$	$(3,7,68)$	$(1,0)$
$2465$	$(1,4,7)$	$(2,0)$		$\mathrm{410\hspace{0.17em}041}$	$(5,9,17)$	$(1,0)$
$6601$	$(3,11,20)$	$(1,0)$		$\mathrm{530\hspace{0.17em}881}$	$(1,8,35)$	$(2,0)$
$8911$	$(1,3,11)$	$(2,0)$		$\mathrm{1\hspace{0.17em}024\hspace{0.17em}651}$	$(1,11,15)$	$(2,0)$
$\mathrm{10\hspace{0.17em}585}$	$(1,7,18)$	$(2,0)$		$\mathrm{1\hspace{0.17em}461\hspace{0.17em}241}$	$(1,2,15)$	$(2,1)$
$\mathrm{15\hspace{0.17em}841}$	$(1,5,12)$	$(2,0)$		$\mathrm{1\hspace{0.17em}615\hspace{0.17em}681}$	$(1,9,16)$	$(2,0)$
$\mathrm{52\hspace{0.17em}633}$	$(1,12,17)$	$(2,0)$		$\mathrm{1\hspace{0.17em}909\hspace{0.17em}001}$	$(2,5,23)$	$(1,0)$
$\mathrm{115\hspace{0.17em}921}$	$(1,3,20)$	$(2,0)$		$\mathrm{2\hspace{0.17em}508\hspace{0.17em}013}$	$(2,3,23)$	$(1,0)$
$\mathrm{162\hspace{0.17em}401}$	$(2,5,29)$	$(1,0)$		$\mathrm{3\hspace{0.17em}057\hspace{0.17em}601}$	$(1,5,8)$	$(2,1)$

Table 11. Table 9.1. Taxicab numbers Ta ​ ( n ) Ta 𝑛 \mathrm{Ta}(n) for n = 3 , … , 6 𝑛 3 … 6 n=3,\dots,6 .

$\mathrm{87\hspace{0.17em}539\hspace{0.17em}319}$$=$	$3^3\cdot 7\cdot 31\cdot 67\cdot 223$
$\mathrm{6\hspace{0.17em}963\hspace{0.17em}472\hspace{0.17em}309\hspace{0.17em}248}$$=$	$2^{10}\cdot 3^3\cdot 7\cdot 13\cdot 19\cdot 31\cdot 37\cdot 127$
$\mathrm{48\hspace{0.17em}988\hspace{0.17em}659\hspace{0.17em}276\hspace{0.17em}962\hspace{0.17em}496}$$=$	$2^6\cdot 3^3\cdot 7^4\cdot 13\cdot 19\cdot 43\cdot 73\cdot 97\cdot 157$
$\mathrm{24\hspace{0.17em}153\hspace{0.17em}319\hspace{0.17em}581\hspace{0.17em}254\hspace{0.17em}312\hspace{0.17em}065\hspace{0.17em}344}$$=$	$2^6\cdot 3^3\cdot 7^4\cdot 13\cdot 19\cdot 43\cdot 73\cdot {79}^3\cdot 97\cdot 157$

Table 12. Table 9.2. Cube-free taxicab numbers Tc ​ ( n ) Tc 𝑛 \mathrm{Tc}(n) for n = 3 , 4 𝑛 3 4 n=3,4 .

$\mathrm{15\hspace{0.17em}170\hspace{0.17em}835\hspace{0.17em}645}$$=$	$3^2\cdot 5\cdot 7\cdot 31\cdot 37\cdot 199\cdot 211$
$\mathrm{1\hspace{0.17em}801\hspace{0.17em}049\hspace{0.17em}058\hspace{0.17em}342\hspace{0.17em}701\hspace{0.17em}083}$$=$	$7\cdot 31\cdot 37\cdot 43\cdot 163\cdot 193\cdot 9151\cdot \mathrm{18\hspace{0.17em}121}$

Equations

$$a^{m-1}\equiv 1(\mathrm{mod}m)$$

$$p\mid m⟹p-1\mid m-1.$$

$$p-1\mid m-1,p-1\mid \frac{m}{p}-1,\text{and}p\nmid q-1.$$

$$\mathcal{C}=\{$$

$$41041,46657,52633,62745,63973,75361,101101,\dots \}.$$

$$s_p(m)\ge p\text{and}{s}_p(m)\equiv 1(\mathrm{mod}p-1).$$

$$p\le \alpha \sqrt{m}\text{with}\alpha =\sqrt{17/33}=0.7177\dots .$$

$${\mathcal{C}}^{\prime }:=\left\{m\in \mathbb{S}:p\mid m⟹s_p(m)=p\right\},$$

$${\mathcal{C}}^{\prime }=\{$$

$$512461,1152271,1193221,1857241,3828001,4335241,\dots \}.$$

$$p\le \alpha \sqrt{m}\text{with}\alpha =\sqrt{66337/132673}=0.7071\dots .$$

$$1729$$

$$10606681$$

$$4872420815346001$$

$$U_n(t)\in \mathbb{Z}[t]\text{ of degree }n\ge 3$$

$$U_3(t)=(6t+1)(12t+1)(18t+1),$$

$$1729$$

$$294409$$

$$56052361$$

$$1729=1^3+12^3=9^3+10^3,$$

$$M$$

$$=1570642921\cdot 3094633081\cdot 7759909081,$$

$$s_p(M)=p$$

$$\frac{z{e}^{xz}}{e^z-1}=\sum _{n\ge 0}{B}_n(x)\frac{z^n}{n!}(|z|<2\pi ).$$

$$B_n(x)-B_n=\sum _{k=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n}{k}\right)B_k{x}^{n-k}.$$

$${\mathbb{D}}_n=\prod _{s_p(n)\ge p}p$$

$$\mathrm{rad}(n+1)\mid {\mathbb{D}}_n,$$

$${\mathbb{D}}_n=\mathrm{lcm}({\mathbb{D}}_{n+1},\mathrm{rad}(n+1)),$$

$$m\in \mathcal{C}⟹m\mid {\mathbb{D}}_m,$$

$${\mathrm{ord}}_g(m):=\max \left\{n\ge 0:g^n\mid m\right\}.$$

$$m=\prod _{\nu =1}^n{g}_{\nu }^{e_{\nu }},$$

$$s_{g_{\nu }}(m)\ge g_{\nu }.$$

$$s_{g_{\nu }}(m)=g_{\nu }.$$

$$\mathfrak{S}$$

Full text

1. Introduction

By Fermat’s little theorem the congruence

[TABLE]

holds for all integers $a$ coprime to $m$, if $m$ is a prime. Moreover, this congruence also holds for positive composite integers $m$, which are called Carmichael numbers and obey the following criterion. Let $p$ always denote a prime.

Theorem 1.1** (Korselt’s criterion [16] (1899)).**

A positive composite integer $m$ is a Carmichael number if and only if $m$ is squarefree and

[TABLE]

Subsequently, Carmichael independently derived further properties of these numbers and computed first examples of them.

Theorem 1.2** (Carmichael [3, 4] (1910,1912)).**

If $m$ is a Carmichael number, then $m$ is a positive odd and squarefree integer having at least three prime factors. Moreover, if $p$ and $q$ are prime divisors of $m$, then

[TABLE]

Denote the set of Carmichael numbers by

[TABLE]

Following [15], the Carmichael numbers can be also characterized in a quite different and surprising way. Let $s_{p}(m)$ be the sum of the base-$p$ digits of $m$.

Theorem 1.3** (Kellner and Sondow [15]).**

An integer $m>1$ is a Carmichael number if and only if $m$ is squarefree and each of its prime divisors $p$ satisfies both

[TABLE]

Moreover, $m$ is odd and has at least three prime factors, each prime factor $p$ obeying the sharp bound

[TABLE]

Define the set of primary Carmichael numbers by

[TABLE]

where $\mathbb{S}=\left\{2,3,5,6,7,10,\dotsc\right\}$ is the set of squarefree integers $m>1$. The first elements are given by

[TABLE]

The set $\mathcal{C}^{\prime}$ (meaning “$\mathcal{C}$ prime”) of primary Carmichael numbers, which was introduced in [15], is indeed a subset of the Carmichael numbers.

Theorem 1.4** (Kellner and Sondow [15]).**

We have $\mathcal{C}^{\prime}\subset\mathcal{C}$. If $m\in\mathcal{C}^{\prime}$, then each prime factor $p$ of $m$ obeys the sharp bound

[TABLE]

We further define for a given set $\mathbf{S}\subseteq\mathcal{C}$ the subsets $\mathbf{S}_{n}\subseteq\mathbf{S}$, where each element of $\mathbf{S}_{n}$ has exactly $n$ prime factors. Let $\mathrm{S}(x)$ and $\mathrm{S}_{n}(x)$ count the number of elements of $\mathbf{S}$ and $\mathbf{S}_{n}$ less than $x$, respectively. We call a squarefree number $m$ with exactly $n$ prime factors briefly an $n$-factor number.

The first element of $\mathcal{C}^{\prime}_{n}$ for $n=3,4,5$ is given by

[TABLE]

respectively.

In 1939 Chernick [5] introduced certain squarefree polynomials

[TABLE]

to construct Carmichael numbers, where $t\geq 0$ is an integer. More precisely, he showed that $U_{n}(t)$ represents a Carmichael number for $t\geq 0$, whenever all $n$ linear factors of $U_{n}(t)$ are simultaneously odd primes. The simplest one of these polynomials is

[TABLE]

which produces the $3$-factor Carmichael numbers

[TABLE]

being the first three examples.

At first glance, one observes that the third-smallest Carmichael number $1729$, which is also known as Ramanujan’s famous taxicab number (being the smallest number that is a sum of two positive cubes in two ways, see Silverman [20]), namely,

[TABLE]

is additionally the smallest primary Carmichael number. Surprisingly, a closer look reveals that the other two numbers $294\,409$ and $56\,052\,361$ are also primary Carmichael numbers. Is this pure coincidence or a hidden phenomenon?

The purpose of this paper is to show that any $U_{3}(t)$ has the property that all values of $U_{3}(t)$ for $t\geq 2$, and apart from certain exceptions also in the case $t=1$, lie in a certain set $\mathfrak{S}^{\prime}$ (as introduced in Section 2) that generalizes the set $\mathcal{C}^{\prime}$.

As a main result of Section 4, it further turns out that any given $U_{3}(t)$ has the following important property: if both $U_{3}(t)\in\mathfrak{S}^{\prime}$ and all three linear factors of $U_{3}(t)$ are odd primes for a fixed $t\geq 0$, then $U_{3}(t)$ represents not only a Carmichael number, but also a primary Carmichael number.

Thus, almost all $3$-factor Carmichael numbers, which were computed by Chernick’s method so far, lie in $\mathcal{C}^{\prime}_{3}$. The restriction “almost” refers to the exceptions in the cases $t=0$ and $t=1$.

As a striking example, in 1980 Wagstaff [22] already computed a very huge $3$-factor Carmichael number with $321$ decimal digits by using $U_{3}(t)$ as defined by (1.1), where $t$ is a $106$-digit number. This number now awakes from a deep sleep as a primary Carmichael number!

In 2002 Dubner [9] also used this $U_{3}(t)$ to compute the corresponding $3$-factor Carmichael numbers up to $10^{42}$, which are all primary.

By this means, one can even find a special $\tilde{U}_{3}(t)$ very quickly such that for $t=1$ the value $M=\tilde{U}_{3}(1)\in\mathcal{C}^{\prime}_{3}$ yields the large example

[TABLE]

satisfying in fact the remarkable property

[TABLE]

for each prime factor $p$ of $M$. The reader is invited to check this property above. See Table 4.4 in Section 4 for the construction.

In 1904 Dickson [8] stated the conjecture that a set of linear functions $f_{\nu}(t)=a_{\nu}t+b_{\nu}\in\mathbb{Z}[t]$, under certain conditions, might be simultaneously prime for infinitely many integral values of $t$.

Hence, Dickson’s conjecture, as already noted by Chernick, implies that any $U_{3}(t)$ produces infinitely many Carmichael numbers, and so the set $\mathcal{C}$ should be infinite. This statement now transfers to the set $\mathcal{C}^{\prime}$ of primary Carmichael numbers.

While the question, whether there exist infinitely many Carmichael numbers, was positively answered by Alford, Granville, and Pomerance [1] in 1994, the related question for the primary Carmichael numbers and their distribution is still open.

Unfortunately, several computations suggest that the properties of $U_{3}(t)$ as described above do not hold for $U_{n}(t)$ with $n\geq 4$. One may speculate whether this causes the high proportion of primary Carmichael numbers with exactly three prime factors among all primary Carmichael numbers, see Table 1.1. However, we raise an explicit conjecture on related properties of $U_{4}(t)$ in Section 4.

Going into more detail, Table 1.1 shows the distributions of $C(x)$, $C^{\prime}(x)$, and their subsets up to $10^{18}$. On the one hand, one observes that in this range about $97\%$ of the primary Carmichael numbers have exactly three factors, the remaining $3\%$ have four and five factors. On the other hand, the ratio $C^{\prime}_{3}(x)/C_{3}(x)$ is steadily increasing for $x$ in the range up to $10^{18}$, implying that about $87\%$ of the $3$-factor Carmichael numbers are primary in that range.

Computed Carmichael numbers and tables up to $10^{18}$ in this paper were taken from Pinch’s tables in [17, 18], while the numbers up to $10^{9}$, in particular for $\mathcal{C}^{\prime}$, were rechecked by our computations. Further tables are given by Granville and Pomerance in [10], which also rely mainly on Pinch’s computations. The used raw data files of [18] are named carmichael-16.gz, carmichael17.gz, carmichael18.gz, and car3-18.gz.

Interestingly, the progress about the (primary) Carmichael numbers, as partially described above, were originally initiated by a completely different context. For the sake of completeness, we give here a short survey of some results of [12, 13, 14, 15].

As usual, denote the Bernoulli polynomials and numbers by $B_{n}(x)$ and $B_{n}=B_{n}(0)$, respectively. The polynomials $B_{n}(x)$ are defined by the series (cf. [6, Sec. 9.1, pp. 3–4])

[TABLE]

Define for $n\geq 1$ the denominators $\mathbb{D}_{n}:=\operatorname{denom}(B_{n}(x)-B_{n})$ of the Bernoulli polynomials, which have no constant term,

[TABLE]

These denominators are given by the notable formula

[TABLE]

and obey several divisibility properties. We have, for example,

[TABLE]

where $\operatorname{rad}(n):=\prod_{p\,\mid\,n}p$. It further turns out that all Carmichael numbers satisfy the divisibility relation

[TABLE]

which explains the unexpected link between Carmichael numbers and the function $s_{p}(\cdot)$.

The rest of the paper is organized as follows. The main results, theorems, and conjectures are presented in Sections 2 – 5 after introducing necessary definitions and complementary results. Subsequently, Sections 6 – 8 contain the proofs of the theorems, ordered by their dependencies. Section 9 shows some connections to the taxicab numbers. Finally, in Section 10 we give applications to the polygonal numbers.

2. Decompositions

Let $\mathbb{N}$ be the set of positive integers. The sum-of-digits function $s_{p}(\cdot)$ is actually defined for any integer base $g\geq 2$ in place of a prime $p$. To avoid ambiguity, we define $s_{1}(m):=0$ for $m\geq 0$. For integers $g\geq 2$ and $m\geq 1$ define

[TABLE]

We say that a positive integer $m$ has an $s$-decomposition, if there exists a decomposition in $n$ proper factors $g_{\nu}$ with exponents $e_{\nu}\geq 1$, the factors $g_{\nu}$ being strictly increasing but not necessarily coprime, such that

[TABLE]

where each factor $g_{\nu}$ satisfies the sum-of-digits condition

[TABLE]

Similarly, we say that (2.1) represents a strict $s$-decomposition, if each factor $g_{\nu}$ satisfies the strict sum-of-digits condition

[TABLE]

Accordingly, we define the sets

[TABLE]

One computes that

[TABLE]

Clearly, we have $\mathfrak{S}^{\prime}\subset\mathfrak{S}$. Some examples of $s$-decompositions are

[TABLE]

Note that an $s$-decomposition of a number $m\in\mathfrak{S}$ does not have to be unique. Such an example of different $s$-decompositions is given by

[TABLE]

showing all possible variants.

While the definition of the set $\mathfrak{S}^{\prime}$ widely extends the definition of the set $\mathcal{C}^{\prime}$, the set $\mathfrak{S}$ widely extends the set

[TABLE]

where

[TABLE]

As introduced and shown in [15], the set $\mathcal{S}$ has the property that $\mathcal{C}\subset\mathcal{S}$. Moreover, each number $m\in\mathcal{S}$ has at least three prime factors.

The next two theorems summarize the properties of $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$, which also show some connections with the Carmichael numbers.

Theorem 2.1**.**

An $s$-decomposition of $m\in\mathfrak{S}$ has the following properties:

1. (i)

The $s$-decomposition of $m$ has at least two factors, while $m$ has at least two prime divisors. 2. (ii)

If $m=g_{1}^{e_{1}}\cdot g_{2}^{e_{2}}$, then $e_{1}+e_{2}\geq 3$. 3. (iii)

If $m=g_{1}\cdot g_{2}\cdot g_{3}$ where all $g_{\nu}$ are odd primes, then its $s$-decomposition is unique. In particular, if $m\in\mathfrak{S}^{\prime}$, then $m\in\mathcal{C}^{\prime}_{3}$. 4. (iv)

If $m=g_{1}\dotsm g_{n}$ with $n\geq 3$, where all $g_{\nu}$ are odd primes, then $m\in\mathcal{S}$. Moreover, if $g_{1}\dotsm g_{n}\in\mathfrak{S}^{\prime}$, then $m\in\mathcal{C}^{\prime}_{n}$. 5. (v)

If $m=g_{1}^{e_{1}}\dotsm g_{n}^{e_{n}}$ with $n\geq 2$, then each factor $g_{\nu}$ satisfies the inequalities $1<g_{\nu}<m^{1/(\operatorname{ord}_{g_{\nu}}(m)+1)}\leq m^{1/(e_{\nu}+1)}\leq\sqrt{m}$.

Theorem 2.2**.**

The sets $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$ have the following properties:

1. (i)

$\mathcal{C}\subset\mathfrak{S}$. 2. (ii)

$\mathcal{C}^{\prime}\subset\mathfrak{S}^{\prime}\,\cap\,\mathcal{C}$. 3. (iii)

$\mathcal{C}^{\prime}_{3}=\mathfrak{S}^{\prime}\,\cap\,\mathcal{C}_{3}$.

We further define the generalized sets of $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$ by

[TABLE]

The sets $\overline{\mathfrak{S}}$ and $\overline{\mathfrak{S}^{\prime}}$ satisfy the conditions (2.2) and (2.3) for at least one proper divisor of each of their elements, respectively. One computes that

[TABLE]

By the definitions and the computed examples we have the relations

[TABLE]

The following two theorems show weaker and different properties of $\overline{\mathfrak{S}}$ and $\overline{\mathfrak{S}^{\prime}}$ compared to Theorems 2.1 and 2.2.

Theorem 2.3**.**

A number $m\in\overline{\mathfrak{S}}$ and a divisor $g\mid m$ with $s_{g}(m)\geq g$ have the following properties:

1. (i)

$m$* has at least two prime divisors.* 2. (ii)

If $m\in\mathcal{C}_{3}$, then $g$ is an odd prime. 3. (iii)

$g$* obeys the inequalities $1<g<m^{1/(\operatorname{ord}_{g}(m)+1)}\leq\sqrt{m}$.*

Theorem 2.4**.**

The set $\overline{\mathfrak{S}^{\prime}}\setminus\mathfrak{S}^{\prime}$ has the following properties:

1. (i)

$\mathcal{C}\setminus\mathcal{C}^{\prime}\not\subset\overline{\mathfrak{S}^{\prime}}\setminus\mathfrak{S}^{\prime}$. 2. (ii)

$\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}\subset\overline{\mathfrak{S}^{\prime}}\setminus\mathfrak{S}^{\prime}$.

Remark**.**

Theorems 2.3 (ii) and 2.4 (ii), and the properties of the set $\mathcal{C}^{\prime}_{3}$ imply that all $3$-factor Carmichael numbers have the following property: every number $m\in\mathcal{C}_{3}$ satisfies the strict sum-of-digits condition (2.3) for at least one prime factor of $m$. This will be stated later more precisely; see Theorems 4.4, 4.5, and 5.2.

If one could show the open question, whether the set $\mathcal{C}^{\prime}$ is infinite, then Theorem 2.2 would imply that $\mathfrak{S}^{\prime}$ is also infinite. Fortunately, the infinitude of $\mathfrak{S}^{\prime}$ can be shown independently of the set $\mathcal{C}^{\prime}$.

Theorem 2.5**.**

The set $\mathfrak{S}^{\prime}$ is infinite.

The relations in (2.4) immediately imply the following corollary.

Corollary 2.6**.**

The sets $\mathfrak{S}$, $\overline{\mathfrak{S}}$, and $\overline{\mathfrak{S}^{\prime}}$ are infinite.

Finally, we define the subsets $\mathfrak{S}_{*}$ and $\mathfrak{S}^{\prime}_{*}$ of the sets $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$, respectively. Each element $m\in\mathfrak{S}_{*}$ (respectively, $m\in\mathfrak{S}^{\prime}_{*}$) has the property that the prime factorization of $m$ equals a (strict) $s$-decomposition. The definitions are given as

[TABLE]

By Theorem 1.3 and the definition of the set $\mathcal{C}^{\prime}$, we have the relations

[TABLE]

While for a given number $m$ the determination of its $s$-decomposition may be difficult due to searching for suitable factors (actually, this problem can be translated into a system of linear equations), the sets $\mathfrak{S}_{*}$ and $\mathfrak{S}^{\prime}_{*}$ can be computed quite easily by checking only prime factorizations. The first numbers that do not have a trivial (strict) $s$-decomposition are given as follows.

[TABLE]

Let $S(x)$ count the number of elements of $\mathfrak{S}$ less than $x$; analogously define this notation for related sets of $\mathfrak{S}$. Table 2.1 shows their distributions compared to $C^{\prime}(x)$ and $C(x)$.

At first glance, a lower bound for the growth of $S^{\prime}(x)$ is given by $O(x^{1/3})$, which will be implied by Theorem 4.4 later. We show this lower bound with explicit and simple constants.

Theorem 2.7**.**

There is the estimate

[TABLE]

3. Exceptional Carmichael Numbers

We introduce the set of exceptional Carmichael numbers by

[TABLE]

By definition we have

[TABLE]

The first numbers in $\mathcal{C}^{\sharp}$ are

[TABLE]

In view of Theorem 2.4, the special properties of the $3$-factor Carmichael numbers can be now restated as follows.

Theorem 3.1**.**

We have $\mathcal{C}^{\sharp}_{3}=\emptyset$.

In the case of the $4$-factor Carmichael numbers, it seems that such exceptions occur very rarely. Indeed, the set $\mathcal{C}^{\sharp}_{4}$ contains only four numbers below $10^{18}$:

[TABLE]

As a consequence of Theorem 1.3, each prime factor $p$ of $m\in\mathcal{C}^{\sharp}$ must satisfy both conditions $s_{p}(m)\geq 2p-1$ and $s_{p}(m)\equiv 1\pmod{p-1}$. Actually, one verifies that the first four numbers $m\in\mathcal{C}^{\sharp}_{4}$, as listed above, even satisfy the condition

[TABLE]

for each prime factor $p$ of $m$.

The $4$-factor Carmichael numbers seem to also play a particular role like the $3$-factor Carmichael numbers. This will be discussed in the next section. Tables 3.1 and 3.2 illustrate the distributions of the sets $\mathcal{C}^{\sharp}_{n}$ and $\mathcal{C}_{n}$, respectively. One also finds Table 3.2 in [10], but with values given up to $10^{16}$.

4. Universal Forms

Chernick [5] introduced so-called universal forms, which are squarefree polynomials in $\mathbb{Z}[t]$, by

[TABLE]

with coefficients $a_{\nu},b_{\nu}\in\mathbb{N}$ satisfying

[TABLE]

for all integers $t\geq 0$ except for the cases when $t=0$ and $b_{\nu}=1$. His results can be summarized as follows.

Theorem 4.1** (Chernick [5] (1939)).**

For each $n\geq 3$ there exist universal forms $U_{n}(t)$ with computable coefficients $a_{\nu},b_{\nu}\in\mathbb{N}$. Moreover, for fixed $n\geq 3$ and $t\geq 0$, a universal form $U_{n}(t)$ represents a Carmichael number in $\mathcal{C}_{n}$, if each factor $a_{\nu}\,t+b_{\nu}$ is an odd prime.

Remark**.**

Chernick required to replace $t$ by $2t$, if all coefficients $a_{\nu}$ and $b_{\nu}$ are odd; otherwise, odd values of $t$ would cause even values of $U_{n}(t)$. Actually, this already happens, if one pair $(a_{\nu},b_{\nu})$ consists of odd integers. However, we explicitly left $t$ unchanged for our purpose. We fix this problem by requiring that a factor $a_{\nu}\,t+b_{\nu}$ must be an odd prime instead of a prime, as stated in Theorems 4.1, 4.2, and 4.4.

For the special case $n=3$ Chernick gave a general construction of $U_{n}(t)$, whereas we use a more suitable formulation by introducing several definitions, as follows.

Define the set

[TABLE]

and the elementary symmetric polynomials for $\mathbf{r}\in\mathcal{R}$ as

[TABLE]

We implicitly use the abbreviation $\sigma_{\nu}$ for $\sigma_{\nu}(\mathbf{r})$, if there is no ambiguity in context. For $\mathbf{r}\in\mathcal{R}$ define the parameter $\ell$ with $0\leq\ell<\sigma_{3}$ satisfying

[TABLE]

One easily verifies the following parity relations for $\mathbf{r}\in\mathcal{R}$.

[TABLE]

Remark**.**

Note that congruence (4.6) is always solvable, since $\sigma_{2}$ is invertible $(\mathrm{mod}\;\sigma_{3})$. This will be shown by Lemma 7.3. Avoiding the expression $1/\sigma_{2}$, Chernick used the compatible expression $\sigma_{2}^{a}\pmod{\sigma_{3}}$ with $a=\varphi(\sigma_{3})-1$, where $\varphi(\cdot)$ is Euler’s totient function.

With the definitions above define the forms with three factors as

[TABLE]

allowing $\mathbf{r}$ as an index in place of $n$.

Theorem 4.2** (Chernick [5] (1939)).**

If $\mathbf{r}\in\mathcal{R}$, then $U_{\mathbf{r}}(t)$ is a universal form. Moreover, for fixed $t\geq 0$, $U_{\mathbf{r}}(t)$ is a Carmichael number in $\mathcal{C}_{3}$, if each of its three factors is an odd prime.

Chernick gave some examples of $U_{\mathbf{r}}(t)$, which are listed in Table 4.2. The simplest one is

[TABLE]

as used in the introduction. The following theorem shows some unique properties of this $U_{\mathbf{r}}(t)$, compared to the case $\mathbf{r}\neq(1,2,3)$.

Theorem 4.3**.**

Let $\mathbf{r}\in\mathcal{R}$ and rewrite (4.10) as

[TABLE]

Then $U_{\mathbf{r}}(t)$ has the following properties for $t\in\mathbb{Z}$:

1. (i)

If $\mathbf{r}=(1,2,3)$, then there are the equivalent properties

[TABLE]

Moreover, one has in this case

[TABLE] 2. (ii)

If $\mathbf{r}\neq(1,2,3)$, then $\ell\neq 0$, $b_{\nu}\neq 1$ $(\nu=1,2,3)$, and

[TABLE]

Otherwise, the parity of $U_{\mathbf{r}}(t)$ alternates. More precisely, if $\sigma_{3}$ is odd, then

[TABLE]

where

[TABLE]

The next theorem shows the following remarkable property of $U_{\mathbf{r}}(t)$. Given any $\mathbf{r}\in\mathcal{R}$ we have that $U_{\mathbf{r}}(t)\in\mathfrak{S}^{\prime}$ for $t\geq 2$. Besides certain exceptions this property also holds in the case $t=1$. More precisely, for those $t\geq 1$ in question the three factors of $U_{\mathbf{r}}(t)$, as given by (4.10), already form a strict $s$-decomposition. If the three factors are odd primes, then $U_{\mathbf{r}}(t)\in\mathcal{C}_{3}$ by Theorem 4.2. Moreover, using the property $U_{\mathbf{r}}(t)\in\mathfrak{S}^{\prime}$, it then follows that $U_{\mathbf{r}}(t)\in\mathcal{C}^{\prime}_{3}$. Thereby we arrive at our main results.

Theorem 4.4**.**

Let $\mathbf{r}\in\mathcal{R}$ and define

[TABLE]

If $t\geq\tau$, then

[TABLE]

where the three factors are given by

[TABLE]

and yield a strict $s$-decomposition. Moreover, if each factor $g_{\nu}$ is an odd prime, then $U_{\mathbf{r}}(t)$ represents a primary Carmichael number, namely,

[TABLE]

The complementary cases omitted by Theorem 4.4 are handled by the following theorem.

Theorem 4.5**.**

Let $\mathbf{r}\in\mathcal{R}$ and the symbols defined as in Theorem 4.4. Define the integer parameter

[TABLE]

For the complementary cases

[TABLE]

the following statements hold.

1. (i)

If each factor $g_{\nu}$ is an odd prime, then $U_{\mathbf{r}}(t)\in\mathcal{C}_{3}$. Additionally,

[TABLE]

In particular, for $m=U_{\mathbf{r}}(t)$ there are the following properties.

1. (ii)

If $t=0$, then

[TABLE] 2. (iii)

If $t=1$, then $m\in\mathfrak{S}$ and its $s$-decomposition $g_{1}\cdot g_{2}\cdot g_{3}\in\mathfrak{S}\setminus\mathfrak{S}^{\prime}$ with $s_{g_{1}}(m)=2g_{1}-1,\,s_{g_{2}}(m)=g_{2},\,s_{g_{3}}(m)=g_{3}$.

Remark**.**

To ensure the property $U_{\mathbf{r}}(t)\in\mathfrak{S}^{\prime}$, the parameter $\tau\in\left\{1,2\right\}$ in Theorem 4.4 cannot be improved in general. Table 4.1 shows examples (taken from Tables 4.2 and 4.3) that satisfy the conditions of Theorem 4.5. Note that for $\mathbf{r}=(1,2,7)$ the decomposition $3\cdot 5\cdot 15\notin\mathfrak{S}$, while the value satisfies $U_{\mathbf{r}}(0)=225=5^{2}\cdot 9\in\mathfrak{S}^{\prime}$. The case $t=0$ and $\vartheta=2$, implying that $U_{\mathbf{r}}(0)$ is a square, is established by a relationship between $U_{\mathbf{r}}(t)$ and the polygonal numbers, see Section 10.

Table 4.2 reproduces the examples of $U_{\mathbf{r}}(t)$ given by Chernick, while we give further examples in Table 4.3. Both tables are extended by a third column with parameters $(\sigma_{1},\sigma_{2},\sigma_{3},\ell,\tau)$.

The example of a special $U_{\mathbf{r}}(1)\in\mathcal{C}^{\prime}_{3}$, which was used in the introduction as $\tilde{U}_{3}(1)$, is shown in Table 4.4. To find such an example, the parameter $\mathbf{r}=(p_{1},p_{2},p_{3})$ was constructed by primes that were chosen from a finite set of primes.

At the end of this section, we consider the case when $U_{n}(t)$ has $n\geq 4$ factors. Unfortunately, several computations suggest that the strong property $U_{n}(t)\in\mathfrak{S}^{\prime}$, which is a necessary (but not sufficient) condition for $U_{n}(t)$ to be in $\mathcal{C}^{\prime}_{n}$, breaks down for $n\geq 4$.

However, it seems that a weaker property, if we replace $\mathfrak{S}^{\prime}$ by $\overline{\mathfrak{S}^{\prime}}\setminus\mathfrak{S}^{\prime}$, still holds in the case $n=4$. This situation may be confirmed by adapting the proof of Theorem 4.4 from case $n=3$ to $n=4$, roughly speaking.

For a provisional verification one can use Chernick’s examples of $U_{4}(t)$ in [5]. On the basis of extended computations and considering the set $\mathcal{C}^{\sharp}_{4}$ of exceptional Carmichael numbers, we raise the following conjecture for the more complicated case $n=4$.

Conjecture 4.6**.**

If $U_{4}(t)$ is a universal form, then $U_{4}(t)$ satisfies the following properties for all sufficiently large $t$:

1. (i)

$U_{4}(t)\in\overline{\mathfrak{S}^{\prime}}\setminus\mathfrak{S}^{\prime}$. 2. (ii)

$U_{4}(t)\notin\mathcal{C}^{\prime}_{4}$.

5. Complementary Cases

Chernick showed that any number $m\in\mathcal{C}_{3}$ obeys a special formula, which is intimately connected with $U_{\mathbf{r}}(t)$. Actually, he defined his universal forms thereafter. Recall the definitions of $\sigma_{\nu}$ and $\ell$ in (4.3) – (4.6). The result can be stated as follows.

Theorem 5.1** (Chernick [5] (1939)).**

If $m\in\mathcal{C}_{3}$, then there exists a unique $\mathbf{r}\in\mathcal{R}$ such that

[TABLE]

where $u$ is an even positive integer. More precisely, if $m=p_{1}\cdot p_{2}\cdot p_{3}$ with odd primes $p_{1}<p_{2}<p_{3}$, then

[TABLE]

Moreover,

[TABLE]

where $t\geq 0$ is an integer satisfying $u=\sigma_{3}\,t+\ell$.

As a result of Theorem 4.4, we have for any $\mathbf{r}\in\mathcal{R}$ that

[TABLE]

where $\tau\in\left\{1,2\right\}$. Moreover,

[TABLE]

when $p_{1}$, $p_{2}$, and $p_{3}$ are odd primes.

In the complementary cases $0\leq t<\tau$, Theorem 4.5 predicts that $U_{\mathbf{r}}(t)\in\mathcal{C}^{\prime}_{3}$ can only happen when $t=0$. Table 5.1 shows the first of those values with parameters $\mathbf{r}$ and $(\tau,t)$.

The remaining values, where $U_{\mathbf{r}}(t)\notin\mathfrak{S}^{\prime}$ for $0\leq t<\tau$, can be viewed as exceptions. The next theorem clarifies these cases in the context of Carmichael numbers $m\in\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}$.

Theorem 5.2**.**

If $m\in\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}$, then we have

[TABLE]

where the greatest prime divisor $p$ of $m$ satisfies

[TABLE]

Moreover, there exist a unique $\mathbf{r}\in\mathcal{R}$, as defined in Theorem 5.1, and an integer $t$ such that

[TABLE]

with $0\leq t<\tau$, where $\tau\in\left\{1,2\right\}$ is defined as in Theorem 4.4.

In the case $(\tau,t)=(2,1)$, property (5.2) also holds for the second greatest prime divisor $p$ of $m$.

Remark**.**

For several numbers $m=p_{1}\cdot p_{2}\cdot p_{3}\in\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}$ in the case $t=0$, property (5.2) holds, as in the case $(\tau,t)=(2,1)$, also for $p_{2}$. However, the first example occurs for

[TABLE]

where (5.2) does not hold for $p_{2}$, as verified by

[TABLE]

The first numbers $m\in\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}$ with parameters $\mathbf{r}$ and $(\tau,t)$ are listed in Table 5.2. By Theorem 5.2 such numbers can be represented by $U_{\mathbf{r}}(t)$ with certain $\mathbf{r}\in\mathcal{R}$ only in the cases $0\leq t\leq 1$, while for any $\mathbf{r}\in\mathcal{R}$ each $U_{\mathbf{r}}(t)$ represents only primary Carmichael numbers for $t\geq 2$ when satisfying (5.1).

Supported by computations of the ratio $C^{\prime}_{3}(x)/C_{3}(x)$ in Table 1.1, Dickson’s conjecture, applied to $U_{\mathbf{r}}(t)$, implies the following conjecture.

Conjecture 5.3**.**

We have

[TABLE]

Due to the very special properties of the primary Carmichael numbers, one may initially believe that these numbers play a minor role when comparing the distributions of $C(x)$ and $C^{\prime}(x)$ in Table 1.1. Only a closer look at the case of $3$-factor Carmichael numbers reveals that primary Carmichael numbers play admittedly a central role in that context.

6. Proofs of Theorems 2.1 and 2.3

Recall the definitions of Section 2.

Lemma 6.1**.**

Let $g,m\in\mathbb{N}$. If $g\mid m$ and $s_{g}(m)\geq g$, then

[TABLE]

Proof.

Since $s_{1}(m)=0$ and $s_{m}(m)=1$, the conditions $g\mid m$ and $s_{g}(m)\geq g$ imply that $g$ is a proper divisor of $m$, and therefore $1<g<m$. Letting $e=\operatorname{ord}_{g}(m)\geq 1$, we can write $m=g^{e}\,m^{\prime}$ with $g\nmid m^{\prime}$. Since $m^{\prime}<g$ would imply $s_{g}(m)=s_{g}(m^{\prime})<g$, it follows that $m^{\prime}>g$. As a consequence, we obtain $m>g^{e+1}\geq g^{2}$, showing the result. ∎

Proof of Theorem 2.1.

Let $m\in\mathfrak{S}$. We have to show five parts.

(i). Since $m=g_{1}^{e_{1}}$ with $e_{1}\geq 1$ yields $s_{g_{1}}(m)=1$, $m$ must have at least two factors in its $s$-decomposition. Next we consider the prime factorization $m=p^{e}$ with $e\geq 1$. For any factor $g=p^{\nu}$ of $m$ with $1\leq\nu\leq e$, we infer that $s_{g}(m)<g$. Thus, $m$ has no $s$-decomposition in this case. Finally, $m$ must have at least two prime factors.

(ii). Assume that $m=g_{1}\cdot g_{2}$ is an $s$-decomposition. With $g_{1}<g_{2}$ we then obtain that $s_{g_{2}}(m)=s_{g_{2}}(g_{1})<g_{2}$, getting a contradiction. This implies that $m=g_{1}^{e_{1}}\cdot g_{2}^{e_{2}}$ must satisfy $e_{1}+e_{2}\geq 3$.

(iii). We have $m=g_{1}\cdot g_{2}\cdot g_{3}$, where all $g_{\nu}$ are odd primes. Assume that the $s$-decomposition of $m$ is not unique. Then by part (i) we would have $m=\tilde{g}_{1}\cdot\tilde{g}_{2}$ as a further $s$-decomposition, where $\tilde{g}_{1}$ is a prime and $\tilde{g}_{2}$ is a product of two primes, or vice versa. But this contradicts part (ii). Additionally, If $m\in\mathfrak{S}^{\prime}$, then $m$ also satisfies the condition to be in $\mathcal{C}^{\prime}$, and thus $m\in\mathcal{C}^{\prime}_{3}$.

(iv). We have the inclusions $\mathcal{S}\subset\mathfrak{S}$ and $\mathfrak{S}^{\prime}\subset\mathfrak{S}$. If $m$ has the $s$-decomposition $g_{1}\dotsm g_{n}$ with $n\geq 3$ factors, where $g_{\nu}$ are odd primes, then $m\in\mathcal{S}$ by definition. Similarly, if $g_{1}\dotsm g_{n}\in\mathfrak{S}^{\prime}$ is a strict $s$-decomposition, then $m\in\mathcal{C}^{\prime}_{n}$.

(v). The exponent $e_{\nu}$ of each factor $g_{\nu}$ of the $s$-decomposition of $m$ satisfies $e_{\nu}\leq\operatorname{ord}_{g_{\nu}}(m)$. The result then follows from Lemma 6.1.

This completes the proof of the theorem. ∎

Proof of Theorem 2.3.

Let $m\in\overline{\mathfrak{S}}$ and $g\mid m$ with $s_{g}(m)\geq g$. We have to show three parts.

(i). Assume that $m=p^{e}$ with $e\geq 1$. Then $g=p^{\nu}$ with $1\leq\nu\leq e$. Since $s_{g}(m)<g$, we get a contradiction. Therefore $m$ must have at least two prime factors.

(ii). We have $m\in\mathcal{C}_{3}\subset\mathfrak{S}$. From Theorems 1.3 and 2.1 (iii), it follows that $m=p_{1}\cdot p_{2}\cdot p_{3}$ is a unique $s$-decomposition, implying that $g$ is an odd prime.

(iii). The inequalities follow from Lemma 6.1, finishing the proof. ∎

7. Proofs of Theorems 4.3, 4.4,

4.5, and 5.2

Let $\mathbb{Z}_{p}$ be the ring of $p$-adic integers, $\mathbb{Q}_{p}$ be the field of $p$-adic numbers, and $\operatorname{v\mspace{-1.5mu}}_{p}(s)$ be the $p$-adic valuation of $s\in\mathbb{Q}_{p}$. As a basic property of $p$-adic numbers, we have

[TABLE]

where equality holds if $\operatorname{v\mspace{-1.5mu}}_{p}(x)\neq\operatorname{v\mspace{-1.5mu}}_{p}(y)$ (see [19, Sec. 1.5, pp. 36–37]).

For $x\in\mathbb{R}$ we write $x=\left[x\right]+\left\{x\right\}$, where $\left[x\right]$ denotes the integer part, and $0\leq\left\{x\right\}<1$ denotes the fractional part. Recall the definitions of $\sigma_{\nu}$ and $\ell$ in (4.3) – (4.6). We set $J:=\left\{1,2,3\right\}$ and use $j\in J$ as an index, mainly in the context of $\mathbf{r}\in\mathcal{R}$. Before proving the theorems, we need several lemmas.

Lemma 7.1**.**

Let $\mathbf{r}\in\mathcal{R}$. If $\mathbf{r}=(1,2,3)$, then $\sigma_{3}=\sigma_{1}=6$; otherwise, $\sigma_{3}>\sigma_{1}>6$.

Proof.

First we consider the triple $(1,2,r_{3})\in\mathcal{R}$ with $r_{3}\geq 3$. We then obtain that $\sigma_{3}=2r_{3}\geq 3+r_{3}=\sigma_{1}\geq 6$, where equality can only hold for $r_{3}=3$, respectively, $(1,2,3)\in\mathcal{R}$. This shows this case. Since $r_{1}<r_{2}<r_{3}$ for $\mathbf{r}\in\mathcal{R}$, there remains the case where $r_{1}r_{2}\geq 3$. It follows that $\sigma_{3}\geq 3r_{3}>3r_{3}-3\geq\sigma_{1}>6$, completing the proof. ∎

Lemma 7.2**.**

Let $\mathbf{r}\in\mathcal{R}$ and $j\in J$. Define

[TABLE]

If $r_{j}\geq 2$, then

[TABLE]

Proof.

Let $r_{j}\geq 2$. One observes by (4.4) and (4.5) that

[TABLE]

since the integers $r_{1}$, $r_{2}$, and $r_{3}$ are pairwise coprime. The congruence $\sigma_{1}\equiv\sigma^{\prime}_{1}\pmod{r_{j}}$ follows from the definition. ∎

Lemma 7.3**.**

Let $\mathbf{r}\in\mathcal{R}$ and the parameter $\ell$ be defined as in (4.6) by

[TABLE]

where $0\leq\ell<\sigma_{3}$. The congruence is always solvable, since $\sigma_{2}$ is invertible $(\mathrm{mod}\;\sigma_{3})$. In particular,

[TABLE]

Proof.

By (7.3) we have for $j\in J$ and $r_{j}\geq 2$ that

[TABLE]

Note that in case $r_{1}=1$ we have to consider $\sigma_{3}=r_{2}r_{3}$ with two factors instead of $\sigma_{3}=r_{1}r_{2}r_{3}$. Since the integers $r_{j}$ are pairwise coprime, it follows that $\sigma_{2}$ is invertible $(\mathrm{mod}\;\sigma_{3})$ by (7.5). Therefore, $\ell=0$ if and only if $\sigma_{1}\equiv 0\pmod{\sigma_{3}}$. As $\sigma_{3}\geq\sigma_{1}>0$ and $\sigma_{3}=\sigma_{1}$ if and only if $\mathbf{r}=(1,2,3)$ by Lemma 7.1, relation (7.4) follows. ∎

Lemma 7.4**.**

If $\mathbf{r}\in\mathcal{R}$ and $j\in J$, then

[TABLE]

is an integer, and the bound is sharp. In particular, $\eta=2$ holds for $j=3$ in both cases $\ell=0$ and $\ell\neq 0$ by $\mathbf{r}=(1,2,3)$ and $\mathbf{r}=(1,2,7)$, respectively.

Proof.

If $r_{j}=1$, then $\eta$ is integral. Assume that $r_{j}\geq 2$. Using Lemmas 7.2 and 7.3, we obtain

[TABLE]

For the reduced numerator of $\eta$ we then infer that

[TABLE]

implying that $\eta$ is integral. For any $r_{j}\geq 1$, we have $\eta\geq\sigma_{1}/r_{j}=1+\sigma^{\prime}_{1}/r_{j}>1$, so $\eta\geq 2$. In particular, one computes $\eta=2$ for $r_{3}$ by taking $\mathbf{r}=(1,2,3)$ and $\mathbf{r}=(1,2,7)$ from Tables 4.2 and 4.3, respectively. Both examples incorporate the cases $\ell=0$ and $\ell\neq 0$. This completes the proof. ∎

Lemma 7.5**.**

Let $\mathbf{r}\in\mathcal{R}$ and $j\in J$ where $r_{j}\geq 2$. Define

[TABLE]

Then $\alpha,\beta,\alpha+\beta\in\mathbb{Z}/r_{j}^{2}\setminus\mathbb{Z}$ are fractions.

Proof.

By (7.2) rewrite $\alpha$ and $\beta$ as

[TABLE]

Obviously, we have $\alpha,\beta,\alpha+\beta\in\mathbb{Z}/r_{j}^{2}$. As $r_{j}\geq 2$, we show that $\alpha,\beta,\alpha+\beta\notin\mathbb{Z}$. Let $p$ be a prime divisor of $r_{j}$ and $e=\operatorname{v\mspace{-1.5mu}}_{p}(r_{j})\geq 1$. Since $\sigma^{\prime}_{3}$ and $r_{j}$ are coprime, it follows that $\operatorname{v\mspace{-1.5mu}}_{p}(\alpha)=-2e<0$ and thus $\alpha\notin\mathbb{Z}$. In the same way, we infer by (7.1) that $\alpha-\sigma^{\prime}_{1}/r_{j}=\beta\notin\mathbb{Z}$, since $\operatorname{v\mspace{-1.5mu}}_{p}(\alpha)<\operatorname{v\mspace{-1.5mu}}_{p}(\sigma^{\prime}_{1}/r_{j})=\operatorname{v\mspace{-1.5mu}}_{p}(\sigma^{\prime}_{1})-e$. Next we consider

[TABLE]

where we distinguish between two cases as follows.

Case $p\geq 3$. From $\operatorname{v\mspace{-1.5mu}}_{p}(2\alpha)=\operatorname{v\mspace{-1.5mu}}_{p}(\alpha)<\operatorname{v\mspace{-1.5mu}}_{p}(\sigma^{\prime}_{1}/r_{j})$ and using (7.1), we derive that $\alpha+\beta\notin\mathbb{Z}$.

Case $p=2$. We have that $r_{j}$ is even. Due to $\mathbf{r}\in\mathcal{R}$ and the $r_{\nu}$ being pairwise coprime, $\sigma^{\prime}_{3}$ and $\sigma^{\prime}_{1}$ must be odd and even, respectively. Hence, $\operatorname{v\mspace{-1.5mu}}_{p}(2\alpha)=1-2e<0$, while $\operatorname{v\mspace{-1.5mu}}_{p}(\sigma^{\prime}_{1}/r_{j})\geq 1-e$. By (7.1) we get $\alpha+\beta\notin\mathbb{Z}$.

This completes the proof. ∎

Lemma 7.6**.**

Let $\mathbf{r}\in\mathcal{R}$ and $j\in J$ where $r_{j}\geq 2$. Let $\alpha$ and $\beta$ be defined as in Lemma 7.5, and $g=r_{j}\,(\sigma_{3}\,t+\ell)+1$ with $t\in\mathbb{Z}$. Define

[TABLE]

There are the following properties:

1. (i)

If $t\in\mathbb{Z}$, then $\theta\in\mathbb{Z}$. 2. (ii)

If $t\geq 1$, then there are the inequalities

[TABLE] 3. (iii)

If $\mathbf{r}\neq(1,2,3)$, $j=3$, and $t=0$, then

[TABLE]

Proof.

We implicitly use the definitions of (7.2) and (7.8). We have to show three parts.

(i). As $r_{j}\geq 2$ and $t\in\mathbb{Z}$, we obtain by (7.7) that

[TABLE]

Since $\alpha=\left[\alpha\right]+\left\{\alpha\right\}$, it suffices to show that $\alpha g-\beta\in\mathbb{Z}$. We then infer that

[TABLE]

For the latter numerator in reduced form, it follows from (7.11) that

[TABLE]

implying that $\theta\in\mathbb{Z}$.

(ii). We consider the inequalities (7.9). First we show for $t\geq 1$ that

[TABLE]

or equivalently that

[TABLE]

Note that $\beta$ can be negative, so this inequality is not trivial. Since by Lemma 7.5 $\alpha\in\mathbb{Z}/r_{j}^{2}\setminus\mathbb{Z}$ is a fraction, we obtain that

[TABLE]

For $t\geq 1$ we have

[TABLE]

Combining both inequalities above, we deduce that

[TABLE]

Therefore, we show the following inequality

[TABLE]

Let $i,k\in J\setminus\left\{j\right\}$ be the other two indices complementary to $j$. Then the above inequality becomes

[TABLE]

Since $r_{i},r_{k}\geq 1$ but $r_{i}\neq r_{k}$, we can use the estimate

[TABLE]

It is easy to see that $\mathfrak{g}(r_{j})$ is strictly decreasing for $r_{j}\geq 2$. Hence, $\mathfrak{g}(2)=3/4>\mathfrak{f}(r_{j})$ for $r_{j}\geq 2$, implying that (7.16) holds for $t\geq 1$. Finally, putting all together yields for $t\geq 1$ that

[TABLE]

Now we show for $t\geq 1$ that

[TABLE]

Since both sides of the above inequality lie in $\mathbb{Z}$, we can also write

[TABLE]

By the same arguments, the inequalities (7.13) and (7.15) are also valid for $\left\{\alpha\right\}$ in place of $1-\left\{\alpha\right\}$. In view of (7.14), we then have

[TABLE]

Hence, we proceed in showing that

[TABLE]

This turns into

[TABLE]

Since $\sigma^{\prime}_{3}\geq 2$ and $r_{j}\geq 2$, we obtain the estimates

[TABLE]

As a consequence, we infer that $S<1$, and thus (7.17) holds for $t\geq 1$. Again, putting all together yields for $t\geq 1$ that

[TABLE]

finally showing the inequalities (7.9).

(iii). We consider the case where $\mathbf{r}\neq(1,2,3)$, $j=3$, and $t=0$. Therefore $r_{j}\geq 4$, and $\ell\geq 1$ by Lemma 7.3. Since $r_{1}<r_{2}<r_{3}$, we have $\alpha=\sigma^{\prime}_{3}/r_{j}^{2}<1$ and so $\left\{\alpha\right\}=\alpha$. By $\alpha g-\beta\in\mathbb{Z}$ the inequalities (7.10) become

[TABLE]

where

[TABLE]

From (7.12) we deduce that

[TABLE]

implying that $\alpha g-\beta\geq 1$. There remains to show that $g-1\geq\alpha g-\beta$. After dividing by $g-1=r_{j}\,\ell$, we obtain

[TABLE]

Since $\ell\geq 1$, we continue with

[TABLE]

From $r_{j}>3$ and using the inequalities

[TABLE]

we infer that

[TABLE]

implying that (7.19) holds and so $g-1\geq\alpha g-\beta$. This finally shows the inequalities (7.10), completing the proof. ∎

Now we are ready to give the proofs of the theorems.

Proof of Theorem 4.3.

Let $\mathbf{r}\in\mathcal{R}$. By (4.10) and (4.12) we consider

[TABLE]

Expanding the first product of (7.20) yields

[TABLE]

We have to show two parts.

(i). Comparing both products of (7.20), we infer that

[TABLE]

and from Lemma 7.3 it follows that $\ell=0$ if and only if $\mathbf{r}=(1,2,3)$.

Now let $\mathbf{r}=(1,2,3)$. We have $(\sigma_{1},\sigma_{2},\sigma_{3})=(6,11,6)$. Since $\ell=0$ and $\sigma_{1}=\sigma_{3}$, we deduce from (7.21) that

[TABLE]

For any $t\in\mathbb{Z}$ we obtain

[TABLE]

This finally implies that

[TABLE]

implying the two claimed congruences. From (7.22) we then derive that

[TABLE]

Thus, $U_{\mathbf{r}}(t)$ is odd for all $t\in\mathbb{Z}$.

(ii). Let $\mathbf{r}\neq(1,2,3)$. Then we have $0<\ell<\sigma_{3}$ and by (7.20) that $b_{j}\neq 1$ $(j\in J)$. Using the substitution $\lambda=\sigma_{3}\,t+\ell\neq 0$ for any $t\in\mathbb{Z}$, we obtain by (7.21) that

[TABLE]

Furthermore, it follows from Lemma 7.3 that

[TABLE]

Hence, we infer that

[TABLE]

where $\lambda=\ell$ if $t=0$, $\lambda=\sigma_{3}+\ell$ if $t=1$, and $\gcd(\sigma_{3},\ell)\mid\lambda$ in any case. This implies the claimed congruences

[TABLE]

If $\sigma_{3}$ is even, then (4.9) implies that $2\mid\ell$, and so $2\mid\lambda$. We then derive from (7.24) that

[TABLE]

and $U_{\mathbf{r}}(t)$ is odd for all $t\in\mathbb{Z}$.

Otherwise, $\sigma_{3}$ is odd. In this case it follows from (4.8) that $\sigma_{1}$ and $\sigma_{2}$ are also odd. With that we infer from (7.23) that

[TABLE]

regardless of the parity of $\lambda$, and therefore valid for all $t\in\mathbb{Z}$. Moreover, (7.23) then implies that

[TABLE]

where $\delta(t)=1$ if $t\equiv\ell$ $(\mathrm{mod}\;2)$, and $\delta(t)=0$ otherwise. This shows the alternating parity of $U_{\mathbf{r}}(t)$. If $\delta(t)=1$, then

[TABLE]

Together with (7.25), since $\sigma_{3}\gcd(\sigma_{3},\ell)$ is odd, we finally achieve

[TABLE]

being compatible with the case $\delta(t)=0$. This completes the proof of the theorem. ∎

Proof of Theorem 4.4.

Let $\mathbf{r}\in\mathcal{R}$ and $t\geq 0$ be an integer. As defined in (4.10), write

[TABLE]

where the three factors are given by

[TABLE]

and $0\leq\ell<\sigma_{3}$ by (4.6).

Theorem 4.2 states that $U_{\mathbf{r}}(t)$ is a universal form. We briefly write

[TABLE]

keeping in mind that $m$ and the $g_{j}$ depend on $t$.

We have to determine an integer $\tau\in\left\{1,2\right\}$ as claimed such that the strict sum-of-digits condition holds for $t\geq\tau$ as follows.

[TABLE]

In this case, the right-hand side of (7.27) provides a strict $s$-decomposition of $m$, and thus

[TABLE]

To find the parameter $\tau$, we will derive some conditions on the parameters $(\sigma_{1},\sigma_{3},\ell)$. To show condition (7.28), we proceed for each fixed $j\in J$ as follows. Let $i,k\in J\setminus\left\{j\right\}$ be the other two indices complementary to $j$. We further write

[TABLE]

noting that

[TABLE]

Our goal is to find an expression for $m^{\prime}$ in terms of $g$. In view of (7.26) we can effectively rewrite $g_{i}$ and $g_{k}$ as

[TABLE]

We then derive initially the expression

[TABLE]

where all terms and fractions still yield integers. Since we need an expansion in $g$, we finally attain to the following expression for $m^{\prime}$ with rational coefficients.

[TABLE]

with

[TABLE]

obeying

[TABLE]

where

[TABLE]

We deduce from Lemma 7.5 that

[TABLE]

The case $r_{j}=1$ can only happen when $j=1$, while the coefficients are integers. In the other case the coefficients are fractions. However, there arises the problem of finding a suitable $g$-adic expansion of (7.31) to show that in fact $s_{g}(m^{\prime})=g$. To proceed in this way, we let “the coefficients $\gamma_{\nu}$ float”. We have to distinguish between the following two cases.

Case $r_{j}=1$. We rewrite (7.31) by (7.32) and (7.33) as

[TABLE]

with the coefficients

[TABLE]

and the parameter $\lambda\in\left\{1,2\right\}$.

Next we show that the integers $a_{\nu}$ are $g$-adic digits, so satisfying

[TABLE]

which implies that

[TABLE]

By Lemma 7.1 we have the inequalities

[TABLE]

and by (7.26) that

[TABLE]

Thus, we infer that (7.35) holds for $a_{0}$ and $a_{2}$, if $t\geq 1$ and $\lambda=1,2$. For $a_{1}$ we first consider (7.34) with $\lambda=1$. The inequalities

[TABLE]

are valid for $t\geq 2$ unconditionally, and for $t=1$ if $\ell\geq\sigma_{3}-\sigma_{1}$. Hence, (7.35) holds for $a_{1}$ in these two cases.

We now consider the remaining case $t=1$ and $\ell<\sigma_{3}-\sigma_{1}$ with $\lambda=2$. From (7.34) and (7.36) we then derive the inequalities

[TABLE]

which are valid by assumption, showing that (7.35) also holds for $a_{1}$ in that case.

Finally, we achieve the conditions for the case $r_{j}=1$ as

[TABLE]

This completes the first case $r_{j}=1$.

Case $r_{j}>1$. We rewrite (7.31) as

[TABLE]

with the coefficients

[TABLE]

One observes that the coefficients $a_{\nu}$ $(\nu=0,1,2)$ are integers. Moreover, they satisfy that

[TABLE]

There remains to show that the coefficients $a_{\nu}$ are in fact proper $g$-adic digits, implying that $s_{g}(m^{\prime})=g$ as desired.

For $a_{2}$ and $t\geq 1$ this easily follows from (7.26) and (7.33) so that

[TABLE]

By Lemma 7.6 (ii) and (7.9), we have for $t\geq 1$ the inequalities

[TABLE]

which finally imply that $a_{0},a_{1}\in\left\{1,\dots,g-1\right\}$. As a result, we conclude in the case $r_{j}>1$ that

[TABLE]

Now we consider the special case $j=3$, $t=0$, and $\mathbf{r}\neq(1,2,3)$. By Theorem 4.3 we have $U_{\mathbf{r}}(t)>1$, $\ell>0$, and $g>1$. Since $r_{1}<r_{2}<r_{3}$, we infer that

[TABLE]

Therefore $\alpha=\left\{\alpha\right\}$ and $\left[\alpha\right]=0$. The coefficients $a_{\nu}$ then become

[TABLE]

We can apply Lemma 7.6 (iii) and obtain by (7.10) the inequalities

[TABLE]

Comparing (7.6) and (7.18) yields

[TABLE]

where $\eta\geq 2$ by Lemma 7.4. If $\theta>1$ or equivalently $\eta>2$, then

[TABLE]

implying that $a_{0},a_{1}\in\left\{1,\dots,g-1\right\}$ and $s_{g}(m^{\prime})=g$. Otherwise, we have the case $\theta=1$ and $\eta=2$. This yields $m^{\prime}=g$ and thus $s_{g}(m^{\prime})=1$. Consequently,

[TABLE]

This completes the second case $r_{j}>1$.

Combining both cases $r_{j}=1$ and $r_{j}>1$ yields that

[TABLE]

As a result, if $t\geq\tau$, then

[TABLE]

If $g_{1}$, $g_{2}$, and $g_{3}$ are odd primes, then $m\in\mathcal{C}^{\prime}_{3}$ by Theorem 2.1 (iii). This finishes the proof of the theorem. ∎

Proof of Theorem 4.5.

We continue seamlessly with the proof of Theorem 4.4 and consider the complementary cases

[TABLE]

We have to show three parts (in order of their dependencies).

(iii). If $(\tau,t)=(2,1)$, then we obtain by (7.37) and (7.38) that

[TABLE]

Thus, $m\in\mathfrak{S}$ and its $s$-decomposition $g_{1}\cdot g_{2}\cdot g_{3}\in\mathfrak{S}\setminus\mathfrak{S}^{\prime}$.

(i). Assume that the factors $g_{\nu}$ are odd primes. Theorem 4.2 shows that $m\in\mathcal{C}_{3}$. If $m\in\mathfrak{S}^{\prime}$, then $m\in\mathcal{C}^{\prime}_{3}$ by Theorem 2.1 (iii). But if $(\tau,t)=(2,1)$, then part (iii) implies that $m\notin\mathcal{C}^{\prime}_{3}$.

(ii). We consider the case $t=0$ and $j=3$. We then have the equality $\vartheta=\eta$ by (7.6). If $\mathbf{r}=(1,2,3)$, then we obtain $m=1$ by (4.11) and $\eta=2$ by Lemma 7.4. Since $s_{1}(m)=0$ by definition and $g_{1}=g_{2}=g_{3}=1$, the result follows. If $\mathbf{r}\neq(1,2,3)$, then the implications follow from (7.39). For $\eta=2$ we have by (7.29) and (7.39) that $m^{\prime}=g_{3}=g_{1}g_{2}$, so $m=g_{3}^{2}$. If $\eta>2$, then $s_{g_{3}}(m)=g_{3}$ by (7.39), and Lemma 6.1 implies that $m>g_{3}^{2}$. This completes the proof of the theorem. ∎

Proof of Theorem 5.2.

Let $m\in\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}$, where

[TABLE]

with odd primes $p_{1}<p_{2}<p_{3}$. Theorem 1.3 implies that $m\in\mathfrak{S}$. From Theorem 2.1 (iii), it follows that

[TABLE]

By Theorem 5.1 there exist unique $\mathbf{r}\in\mathcal{R}$ and $t\geq 0$ such that

[TABLE]

while Theorem 4.4 implies that $0\leq t<\tau$ with some $\tau\in\left\{1,2\right\}$, since $m\notin\mathfrak{S}^{\prime}$. Next we consider two cases as follows.

Case $t=0$. Since $m\in\mathfrak{S}$ is no square, we infer from Theorem 4.5 (ii) that (5.2) holds for $p_{3}$.

Case $t=1$. From Theorem 4.5 (iii), it follows that (5.2) holds for $p_{2}$ and $p_{3}$.

Hence, both cases imply that $m\in\overline{\mathfrak{S}^{\prime}}$. This finally yields $m\in(\mathfrak{S}\cap\overline{\mathfrak{S}^{\prime}})\setminus\mathfrak{S}^{\prime}$, showing the result. ∎

8. Proofs of Theorems 2.2, 2.4,

2.5, 2.7, and 3.1

The remaining proofs are given in this section, since they depend on Theorems 4.4 and 5.2. Recall the definitions of Sections 2 and 3. In the following we use the notation $m=p_{1}\dotsm p_{n}$, which means that $p_{1}<\dotsb<p_{n}$ are odd primes.

Proof of Theorem 2.2.

We have to show three parts.

(i). Theorem 1.3 implies that $\mathcal{C}\subset\mathfrak{S}$ by definition.

(ii). First we show that $\mathcal{C}^{\prime}\subseteq\mathfrak{S}^{\prime}\cap\mathcal{C}$. If $m\in\mathcal{C}^{\prime}\subset\mathcal{C}$, then $m$ is squarefree and $m=p_{1}\dotsb p_{n}$ with $n\geq 3$, which is a strict $s$-decomposition by definition of $\mathcal{C}^{\prime}$. Thus, $m\in\mathfrak{S}^{\prime}\cap\mathcal{C}$. Next we show that $\mathcal{C}^{\prime}\neq\mathfrak{S}^{\prime}\cap\mathcal{C}$. We search for a counterexample by constructing numbers lying in $\mathfrak{S}^{\prime}$. To do so, we consider as in (4.11) again

[TABLE]

As a result of Theorem 4.4, we have that

[TABLE]

We then find $m=U_{\mathbf{r}}(5)$ with its strict $s$-decomposition and prime factorization as

[TABLE]

One verifies by Korselt’s criterion that $m\in\mathcal{C}$. But since $s_{7}(m)=19$, $m$ fails to be in $\mathcal{C}^{\prime}$. This finally implies that $\mathcal{C}^{\prime}\subset\mathfrak{S}^{\prime}\cap\mathcal{C}$.

(iii). If $m\in\mathcal{C}^{\prime}_{3}\subset\mathcal{C}_{3}$, then $m=p_{1}\cdot p_{2}\cdot p_{3}$ is also a strict $s$-decomposition. Therefore, $m\in\mathfrak{S}^{\prime}\cap\mathcal{C}_{3}$. Contrary, if $m\in\mathfrak{S}^{\prime}\cap\mathcal{C}_{3}$, then $m\in\mathcal{C}^{\prime}_{3}$ by Theorem 2.1 (iii). It follows that $\mathcal{C}^{\prime}_{3}=\mathfrak{S}^{\prime}\cap\mathcal{C}_{3}$. This finishes the proof of the theorem. ∎

Proof of Theorem 2.4.

We have to show two parts.

(i). By definition we have $\mathcal{C}^{\sharp}\subseteq\mathcal{C}\setminus\mathcal{C}^{\prime}$. We use the first example of $\mathcal{C}^{\sharp}_{4}$, namely,

[TABLE]

We have $14$ proper divisors of $m$ (excluding $1$ and $m$). By construction of $\mathcal{C}^{\sharp}$ we have $s_{p}(m)\neq p$ for each prime divisor $p\mid m$. A computational check (e.g., with Mathematica) of the remaining ten proper divisors $g\mid m$ shows each time that $s_{g}(m)\neq g$ , so $m\notin\overline{\mathfrak{S}^{\prime}}$. Finally, it follows that $\mathcal{C}\setminus\mathcal{C}^{\prime}\not\subset\overline{\mathfrak{S}^{\prime}}\setminus\mathfrak{S}^{\prime}$.

(ii). By Theorem 5.2 we have $\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}\subseteq\overline{\mathfrak{S}^{\prime}}\setminus\mathfrak{S}^{\prime}$. Considering the computed examples with only two prime factors, we find that, for example, $6\in\overline{\mathfrak{S}^{\prime}}\setminus\mathfrak{S}^{\prime}$, while $6\notin\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}$. It follows that $\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}\subset\overline{\mathfrak{S}^{\prime}}\setminus\mathfrak{S}^{\prime}$.

This completes the proof of the theorem. ∎

Proof of Theorem 2.5.

We have to show that $\mathfrak{S}^{\prime}$ is infinite. It suffices to use the example in (8.1). By Theorem 4.4 and (8.2), this already implies that infinitely many values of $U_{\mathbf{r}}(t)$, being strictly increasing for $t\geq 1$, lie in $\mathfrak{S}^{\prime}$. ∎

Proof of Theorem 2.7.

Define the real-valued function and its inverse for $x,y\in\mathbb{R}_{\geq 0}$ by

[TABLE]

We have to show that

[TABLE]

While $f(x)$ is strictly increasing for $x\geq 0$, the function $S^{\prime}(x)$ increases stepwise, counting elements of $\mathfrak{S}^{\prime}$ less than $x$. Considering the first values of $\mathfrak{S}^{\prime}=\left\{45,96,\dotsc\right\}$, we have

[TABLE]

From

[TABLE]

we infer that (8.3) holds for $x\in[1,3154]$. By Theorem 4.4 and relations (8.1) and (8.2) we have

[TABLE]

Note that $f^{-1}(y)>g(y)$ for $y\geq 0$, as verified by

[TABLE]

Since $S^{\prime}(x)$ increases after each $x=g(t)$ for $t\in\mathbb{N}$ and $S^{\prime}(97)=2$, we conclude for $x>g(1)=1729$ that

[TABLE]

Combining both intervals for $x$ shows (8.3) and the result. ∎

Proof of Theorem 3.1.

By definition we have $\mathcal{C}^{\sharp}_{3}\subseteq\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}$. Let $m=p_{1}\cdot p_{2}\cdot p_{3}\in\mathcal{C}_{3}\setminus\mathcal{C}^{\prime}_{3}$. Theorem 5.2 shows that $s_{p_{3}}(m)=p_{3}$, implying that $m\notin\mathcal{C}^{\sharp}_{3}$. As a consequence, we infer that $\mathcal{C}^{\sharp}_{3}=\emptyset$. This proves the theorem. ∎

9. Taxicab Numbers

As noted in (1.2), the smallest number which can be written as the sum of two positive cubes in two ways is the number $1729$, known as Ramanujan’s taxicab number or the Hardy–Ramanujan number.

By Section 2 we have the relations

[TABLE]

The $n$th taxicab number $\mathrm{Ta}(n)$ is defined to be the smallest number which can be written as the sum of two positive cubes in $n$ ways. The next numbers $\mathrm{Ta}(n)$ for $n=3,4$ were listed by Silverman [20]. Subsequently, Wilson [23] found $\mathrm{Ta}(5)$, while C. S. Calude, E. Calude, and Dinneen [2] and Hollerbach [11] announced $\mathrm{Ta}(6)$ (see also OEIS [21, Seq. A011541]). Table 9.1 reports these numbers.

Similarly, allowing only cube-free numbers, one finds in [20] and [21, Seq. A080642] the corresponding taxicab numbers $\mathrm{Tc}(n)$ for $n=3,4$, as listed in Table 9.2.

A quick computational check reveals that all taxicab numbers of Tables 9.1 and 9.2 have a common property that

[TABLE]

Therefore, one may raise the following question.

Question**.**

Is there a link between the sets $\mathfrak{S}_{*}$, $\mathfrak{S}^{\prime}_{*}$ and certain integral solutions of the elliptic curve $X^{3}+Y^{3}=A$?

10. Polygonal Numbers

The polygonal numbers (cf. [7, pp. 38–42]) can be defined as follows. For any integer $h\geq 1$, define an $h$-gonal number by

[TABLE]

Special cases are, e.g., the triangular numbers

[TABLE]

while $\mathbf{G}^{4}_{n}=n^{2}$ are the squares, and $\mathbf{G}^{2}_{n}=\mathbf{G}^{n}_{2}=n$ give the trivial cases. For $h=1$ there are only the special cases $\mathbf{G}^{1}_{1}=\mathbf{G}^{1}_{2}=1$; otherwise, $\mathbf{G}^{1}_{n}\leq 0$ for $n\geq 3$.

Recall the definition of a universal form $U_{\mathbf{r}}(t)$ in (4.10), as well as the definitions of $\sigma_{\nu}$ and $\ell$ in (4.3) – (4.6). We further use the definitions and results of Section 7.

The following theorem shows that for any given $\mathbf{r}\in\mathcal{R}$ all values of $U_{\mathbf{r}}(t)$ for $t\geq 0$ are polygonal numbers.

Theorem 10.1**.**

Let $\mathbf{r}\in\mathcal{R}$ and

[TABLE]

where

[TABLE]

Then we have for $t\geq 0$ and $\nu=1,2,3$ the relations

[TABLE]

where $c_{\nu}$ and $d_{\nu}$ are positive integers given by

[TABLE]

In particular,

[TABLE]

Proof.

Set $J=\left\{1,2,3\right\}$ and fix $j\in J$. Let $i,k\in J\setminus\left\{j\right\}$ with $i\neq k$. We solve for $h$ with $g=g_{j}$ the equation

[TABLE]

After some simplifications the equation turns into

[TABLE]

From (7.29) and (7.30), we derive that

[TABLE]

Thus,

[TABLE]

Lemma 7.4 shows that $c_{j}\geq 2$ is a positive integer. Since $r_{j}\mid\sigma_{3}$ and $\sigma_{3}\geq 6$ by Lemma 7.1, we infer that $\sigma_{3}/r_{j}\geq 2$ and so $d_{j}\geq 4$. With $h_{j}=2h$ and $g_{j}=g$, the result follows from (10.1). In particular, we then obtain for $t=0$ and $t\geq 1$ the estimates $h_{j}\geq 4$ and $h_{j}\geq 12$, respectively. This completes the proof of the theorem. ∎

Corollary 10.2**.**

All $3$-factor Carmichael numbers are polygonal numbers. More precisely, if $m\in\mathcal{C}_{3}$, then for each prime divisor $p$ of $m$ there exists a computable integer $h\geq 6$ such that

[TABLE]

Proof.

Let $m\in\mathcal{C}_{3}$. By Theorem 5.1 there exist $\mathbf{r}\in\mathcal{R}$ and $t\geq 0$ such that $m=p_{1}\cdot p_{2}\cdot p_{3}=U_{\mathbf{r}}(t)$. Fix $j\in\left\{1,2,3\right\}$ and set $p=p_{j}$. Applying Theorem 10.1 yields $m=\mathbf{G}^{h}_{p}$ with a computable even integer $h\geq 4$. Since $\mathbf{G}^{4}_{p}=p^{2}$, the case $h=4$ cannot occur, so we finally infer that $h\geq 6$. ∎

We can go further into this connection between polygonal numbers, universal forms, and Carmichael numbers. Considering the factors $g_{\nu}$ of a number $m$ instead of its parametric representation $m=U_{\mathbf{r}}(t)$ leads to a more general result. The following identity explains this elementary relationship in the context of Korselt’s criterion.

Theorem 10.3**.**

We have the identity

[TABLE]

For $g,m\in\mathbb{N}$ and $g\neq 1$, the identity holds if $h\geq 1$ is integral. There are the following statements:

1. (i)

The trivial cases are

[TABLE] 2. (ii)

If $m$ is a Carmichael number and $g$ is a prime divisor of $m$, then identity (10.2) holds where $h\geq 6$ is even. 3. (iii)

For $n\geq 3$ let $U_{n}(t)=g_{1}\dotsm g_{n}$ be a universal form as defined in (4.1), where $g_{\nu}=a_{\nu}\,t+b_{\nu}$ $(1\leq\nu\leq n)$. For fixed $\nu$ and $t\geq 0$, let $m=U_{n}(t)$ where $m>g=g_{\nu}>1$. Then identity (10.2) holds where $h\geq 4$ is even.

Proof.

It is easy to verify that the expression $\mathbf{G}^{h}_{g}$ in (10.2) simplifies to $m$. Let $g,m\in\mathbb{N}$ where $g\neq 1$. Since

[TABLE]

it follows that $h>0$. If $h$ is integral, then $h\geq 1$ and (10.2) holds. We have to show three parts.

(i). Let $g>1$. We infer that

[TABLE]

showing the first equivalence. Let $m\geq 1$. If $g=2$, then $h=m$. Conversely, $h=m$ implies the equation $m=2((m/g-1)/(g-1)+1)$ with solution $g=2$. This shows the second equivalence.

(ii). Let $m\in\mathcal{C}$ and $g\mid m$ be a prime divisor. From Korselt’s criterion it follows that

[TABLE]

Since $m>g>1$, it follows that $d\in\mathbb{N}$. The case $d=1$ would imply $m=g^{2}$, contradicting that $m$ is squarefree. Finally, this implies that $h\geq 6$ is integral and even, showing that (10.2) holds.

(iii). By (4.2) a universal form $U_{n}(t)$ for $n\geq 3$ satisfies

[TABLE]

whenever $g_{\nu}>1$. For fixed $t\geq 0$, $m=U_{n}(t)$, and $g=g_{\nu}>1$, congruence (10.3) follows from $g\mid m$. As $m>g$, we infer that $h\geq 4$ is integral and even, implying that (10.2) holds. This completes the proof of the theorem. ∎

The following example demonstrates the interplay of the preceding results.

Example 10.4**.**

Interestingly, the parameter

[TABLE]

in Theorem 1.4 (note that $132\,673=181\cdot 733$) depends on the number

[TABLE]

which is the least hexagonal number $\mathbf{H}_{p}$ in $\mathcal{C}^{\prime}$ (see [15]). Since $m\in\mathcal{C}^{\prime}_{3}$, Theorem 10.1 furthermore implies that

[TABLE]

for some $\mathbf{r}\in\mathcal{R}$. Indeed, by Theorem 5.1 one finds $\mathbf{r}=(15,61,5528)$, $\sigma_{1}=5604$, $\sigma_{3}=5\,058\,120$, and $\ell=12$. A computation verifies that

[TABLE]

while Theorem 10.3 shows in another way that

[TABLE]

As a final application of Theorem 10.1, we obtain the following result for the taxicab number $1729$.

Example 10.5**.**

Let $\mathbf{r}=(1,2,3)\in\mathcal{R}$. We have $\sigma_{1}=\sigma_{3}=6$ and $\ell=0$ by Table 4.2. Theorem 10.1 provides the relations

[TABLE]

for

[TABLE]

Since $U_{\mathbf{r}}(1)=1729$, we obtain the unified formula

[TABLE]

for

[TABLE]

which yields at once the known relations

[TABLE]