SanD primes and numbers

TL;DR

This paper introduces SanD numbers and SanD primes, analyzing their distribution and asymptotic growth rates in decimal and binary systems, revealing their quasi-fractal digital-sum properties.

Contribution

It defines SanD numbers and primes, and derives their asymptotic growth rates, highlighting their unique digital-sum based structure and distribution patterns.

Findings

SanD numbers grow linearly with constant 2/3

SanD primes grow approximately as x / log^2 x with constant 3/4

Digital-sum function exhibits quasi-fractal behavior affecting convergence

Abstract

We define S(um)anD(ifference) numbers as ordered pairs $(m,\, m+\Delta)$ such that the digital-sum $DS(m(m+\Delta))=\Delta.$ We consider both the decimal and the binary case. If both $m$ and $m+\Delta$ are prime numbers, we refer to SanD {\em primes}. We show that the number of (decimal-based) SanD numbers less than $x$ grows as $c1\cdot x,$ where $c1 = 2/3,$ while the number of SanD primes less than $x$ grows as $c2\cdot x/\log^2{x},$ where $c2 = 3/4.$ Due to the quasi-fractal nature of the digital-sum function, convergence is both slow and erratic compared to twin primes, which, apart from the constant, have the same leading asymptotics.

Tables

Table 1. Table 1: SanD primes data. The contribution from Δ = 5 Δ 5 \Delta=5 adds 1 to each row and is not shown here.

$x$	$\Delta =14$	32	50	68	86	104	122	140	158	176	194
${10}^2$	7	0	0	0	0	0	0	0	0	0	0
$3\cdot {10}^2$	9	4	0	0	0	0	0	0	0	0	0
${10}^3$	11	10	0	0	0	0	0	0	0	0	0
$3\cdot {10}^3$	14	29	1	0	0	0	0	0	0	0	0
${10}^4$	15	69	21	0	0	0	0	0	0	0	0
$3\cdot {10}^4$	16	136	109	2	0	0	0	0	0	0	0
${10}^5$	16	218	464	14	0	0	0	0	0	0	0
$3\cdot {10}^5$	18	329	1310	134	0	0	0	0	0	0	0
${10}^6$	18	451	3579	954	8	0	0	0	0	0	0
$3\cdot {10}^6$	19	582	7740	4099	98	0	0	0	0	0	0
${10}^7$	19	722	15662	16417	1170	2	0	0	0	0	0
$3\cdot {10}^7$	19	826	27871	48714	7831	82	0	0	0	0	0
${10}^8$	19	944	47206	139196	48831	1985	6	0	0	0	0
$3\cdot {10}^8$	19	1014	72994	315414	200810	16247	126	0	0	0	0
${10}^9$	19	1094	106919	696450	813091	135580	3213	0	0	0	0
$3\cdot {10}^9$	19	1134	147652	1347257	2508310	699799	31654	88	0	0	0
${10}^{10}$	19	1178	195617	2499225	7575349	3686127	329134	3302	0	0	0
$3\cdot {10}^{10}$	19	1201	247383	4213080	18918254	13982418	1995357	43223	158	0	0
${10}^{11}$	19	1222	303418	6850021	46040607	53629221	12799997	651464	965	0	0
$3\cdot {10}^{11}$	19	1240	359059	10361558	97588868	163082279	56956080	5104309	18913	8	0
${10}^{12}$	19	1247	414440	15154071	201275729	497036770	264337125	44101608	425673	911	0
$3\cdot {10}^{12}$	19	1262	466029	20993451	373934734	1273600647	938235422	243895420	4365872	21996	3

Table 2. Table 2: Decimal SanD prime analysis. π ​ ( x ) 𝜋 𝑥 \pi(x) is the number of primes < x . absent 𝑥 <x. The totals include the contribution of 1 from Δ = 5 . Δ 5 \Delta=5.

$x$	Total=$T\left(x\right)$	$xT\left(x\right)/\pi {\left(x\right)}^2$	$T\left(x\right){\log }^2\left(x\right)/x$	$T\left(x\right)/{\mathrm{Li}}_2\left(x\right)$
${10}^2$	8	1.2800	1.697	0.7804
$3\cdot {10}^2$	14	1.0926	1.518	0.7965
${10}^3$	22	0.7795	1.050	0.6343
$3\cdot {10}^3$	45	0.7301	0.9615	0.6438
${10}^4$	106	0.7018	0.8992	0.6533
$3\cdot {10}^4$	264	0.7521	0.9352	0.7161
${10}^5$	713	0.7749	0.9450	0.7539
$3\cdot {10}^5$	1792	0.7954	0.9501	0.7789
${10}^6$	5011	0.8132	0.9564	0.8021
$3\cdot {10}^6$	12539	0.8002	0.9297	0.7926
${10}^7$	33993	0.7697	0.8831	0.7639
$3\cdot {10}^7$	85344	0.7418	0.8432	0.7375
${10}^8$	238188	0.7085	0.8082	0.7141
$3\cdot {10}^8$	606625	0.6890	0.7704	0.6862
${10}^9$	1756367	0.6793	0.7543	0.6770
$3\cdot {10}^9$	4735914	0.6809	0.7517	0.6789
${10}^{10}$	14289952	0.6901	0.7576	0.6883
$3\cdot {10}^{10}$	39400953	0.6994	0.7643	0.6978
${10}^{11}$	120276935	0.7092	0.7716	0.7078
$3\cdot {10}^{11}$	333472334	0.7162	0.7763	0.7149
${10}^{12}$	1022747594	0.7231	0.7808	0.7219
$3\cdot {10}^{12}$	2855514856	0.7298	0.7856	0.7287

Table 3. Table 3: Binary SanD prime analysis. π ​ ( x ) 𝜋 𝑥 \pi(x) is the number of primes < x . absent 𝑥 <x.

$x$	Total=$B\left(x\right)$	$xB\left(x\right)/\pi {\left(x\right)}^2$	$B\left(x\right){\log }^2\left(x\right)/x$	$B\left(x\right)/{\mathrm{Li}}_2\left(x\right)$
${10}^2$	6	0.9600	1.2724	0.5853
${10}^3$	32	1.1338	1.5269	0.9226
${10}^4$	172	1.1387	1.4591	1.0601
${10}^5$	922	1.0021	1.2221	0.9749
${10}^6$	5632	0.9140	1.0750	0.9016
${10}^7$	41421	0.9378	1.0761	0.9308
${10}^8$	335551	1.0109	1.1386	1.0061
${10}^9$	2637661	1.0202	1.1328	1.0167
$3\cdot {10}^9$	7017793	1.0090	1.1139	1.0060
${10}^{10}$	20619112	0.9957	1.0932	0.9932
$3\cdot {10}^{10}$	55563472	0.9863	1.0779	0.9840
${10}^{11}$	167019412	0.9849	1.0715	0.9828
$3\cdot {10}^{11}$	460924135	0.9900	1.0730	0.9881
${10}^{12}$	1410277428	0.9970	1.0767	0.9954
$3\cdot {10}^{12}$	3905976118	0.9983	1.0747	0.9968

Table 4. Table 4: Low-order SanD primes.

$\Delta =14$	$\Delta =32$	$\Delta =50$	$\Delta =68$	$\Delta =86$
5	149	2543	19961	412253
17	179	3137	28211	547661
23	239	3407	43541	871163
29	281	4973	44111	937661
53	389	5147	62861	982703
59	431	5693	66821	989381
83	491	7193	69941	992363
113	509	7523	83621	996551
167	569	7649	86561	999917
383	659	7673	88721	999953
443	1019	8243	89261	1296101
1103	1031	8513	92111	1297601
1409	1061	8573	94781	1329863
2003	1259	8627	99191	1336253
3203	1289	9293	120671	1337813
11483	1427	9461	125261	1378253
100043	1439	9497	129461	1410203
200003	1901	9767	129959	1608611
1001003	2081	9833	130211	1642211

Equations

$$s_b(n)=n-(b-1)\sum _{j\ge 1}\lfloor \frac{n}{b^j}\rfloor .$$

$$\mathbb{E}(X_n)\sim \frac{b-1}{2}{\log }_{b}n.$$

$$\mathbb{E}(X_n)-\frac{b-1}{2}{\log }_{b}n=F_1({\log }_{b}n),$$

$$\sum _{n\ge 0}{s}_b(n)z^n=\frac{1}{1-z}\sum _{m\ge 0}\frac{z^{b^m}-b{z}^{b^{m+1}}+(b-1)z^{(b+1)b^m}}{(1-z^{b^m})(1-z^{b^{m+1}})}.$$

$$n=\sum _k{\alpha }_k\cdot 10^k.$$

$$P(a,b)=\frac{1}{\sqrt{2\pi V}}\exp \left(\frac{-(b-a-M{)}^2}{2V}\right).$$

$$(a,b)\equiv (0,0)(\mathrm{mod}3)\mathrm{or}(2,1)(\mathrm{mod}3).$$

$$C_2=\prod _{p\ge 3prime}\left(1-\frac{1}{(p-1{)}^2}\right),$$

$$r=N_1/N_2.$$

$$N_2\sim (1/2)(x/\log x{)}^2,$$

$$\frac{1}{2}/\frac{4}{9}=\frac{9}{8}.$$

$$r=\frac{9T_1}{8T_2},$$

$$T_1\sim \frac{2}{3}x,$$

$$T_2\sim \frac{x^2}{2},\mathrm{so}r=\frac{3}{2x}.$$

$$N_1\sim \frac{3}{2}\cdot \frac{1}{2}\frac{x}{{\log }^{2}x}=\frac{3x}{4{\log }^{2}x}.$$

$$c_b=\prod _q\frac{q(q-2)}{(q-1{)}^2}=\prod _q\left(1-\frac{1}{(q-1{)}^2}\right),$$

$$r=(T_1/T_2)(q-2)/(q-1)/((q-1)/q{)}^2.$$

$$S=3+10^r+10^s,$$

$$S(S+14)=51+2.10^{r+1}+2.10^{s+1}+10^{2r}+10^{2s}+2.10^{r+s}.$$

$${\mathrm{Li}}_2(x)=\frac{x}{{\log }^{2}x}\left(1+\frac{2}{\log x}+\frac{6}{{\log }^{3}x}+O\left(\frac{1}{{\log }^{4}x}\right)\right),$$

$$\frac{{\pi }^2(x)}{x}=\frac{x}{{\log }^{2}x}\left(1+\frac{2}{\log x}+\frac{5}{{\log }^{3}x}+O\left(\frac{1}{{\log }^{4}x}\right)\right),$$

$$W(u)=\frac{1}{\sqrt{2\pi V}}\text{textdagger}\sum _{j\ge 0}\exp \left(-162\frac{{\left(j-\frac{4+M}{18}\right)}^2}{V}\right),$$

$$W(u)=\sum _{j=-\infty }^{j=\infty }\exp \left(-\frac{V{\pi }^2{j}^2}{162}+\pi .ij\frac{4+M}{9}\right).$$

$$W(u)=1+2u^{-a}\cos \left(\frac{\pi }{2}\left({\log }_{10}(u)+\frac{8}{9}\right)\right),$$

$$a=\frac{11.{\pi }^2}{216.\log (10)}\approx \mathrm{0.218.}$$

$$d(n)=$$

$$217,267,334,430,549,715,902,1143,1442,1782,$$

$$P(u)=\frac{1}{12\sqrt{u}}\left(1+2u^{-a}\cos \left[\frac{\pi }{2}\left({\log }_{10}(u)+\frac{8}{9}\right)\right]\right),$$

$$d^{\prime }(n)\equiv \frac{d(n+1)-d(n-1)}{10^{(n+1)/5}-10^{(n-1)/5}}.$$

$$P_{fluc}(u)=P(u)-1/(12\sqrt{u})$$

$$D_{fluc}(u)=d^{\prime }(n)-1/(12\sqrt{10^{n/5}}).$$

Full text

Based on probabilistic arguments, we conjecture that the number of (base-10) SanD numbers less than $x$ grows as $c_{1}\cdot x,$ where $c_{1}=2/3,$ while the number of (base-10) SanD primes less than $x$ grows as $c_{2}\cdot x/\log^{2}{x},$ where $c_{2}=3/4.$

We calculate the number of SanD primes up to $3\cdot 10^{12},$ and use this data to investigate the convergence of estimators of the constant $c_{2}$ to the calculated value. Due to the quasi-fractal nature of the digital-sum function, convergence is both slow and erratic compared to the corresponding calculation for twin primes, though the numerical results are consistent with the calculated results.

AMS Classification scheme numbers: 11A41, 11A63, 11Y55, 11Y60

Key-words: SanD numbers, constrained prime pairs, digital sums, asymptotics of primes.

1 Introduction

In honour of the 95th birthday of one of the authors (FJD), another of the authors (NEF) coined the SanD prime problem. S(um)anD(ifference) primes are defined to be the subset of primes $p,q\in\mathtt{PRIMES}$ with the property that $p\cdot q=r,$ where the sum of the (decimal) digits of $r,$ denoted $s_{10}(r),$ is equal to $q-p=\Delta>0.$

There is only one pair involving the prime 2, viz. $(2,7),$ as $2\cdot 7=14,$ and $s_{10}(14)=7-2=5.$ The next example is $(5,19),$ as $5\cdot 19=95,$ both 5 and 19 are primes and $s_{10}(95)=14=19-5.$ If we relax the requirement of primality, we refer to SanD numbers.

Of course the SanD numbers and SanD primes can be defined in terms of the digital sum in any base $b,$ though $b$ must be even for there to be a non-zero set of such numbers/primes (see Section 4). Here we treat the decimal ($b=10$) and binary ($b=2$) bases in detail, and the general case more superficially. The effect of the digital sum constraint is more prominent in the decimal case.

The study of digital sums goes back at least to Legendre [11]. In the late 18th century he proved that

[TABLE]

Because of the irregular nature of this function, attention historically turned instead to the behaviour of the random variable $s_{b}(U_{n}),$ where $U_{n}$ assumes each of the values $\{0,\ldots,n-1\}$ with equal probability $1/n.$ Let $X_{n}=X_{n}(b)$ denote the random variable $s_{b}(U_{n})$ just defined. The first asymptotic result was proved by Bush [2] in 1940, who showed that

[TABLE]

Mirsky [14] in 1949 showed that the error term in this expression is $O(1),$ a result implicit in Bush’s calculation. A significant improvement was made by Delange [6] who showed that

[TABLE]

where $F_{1}(x)=F_{1}(x+1)$ is a continuous, periodic nowhere differentiable function. An elegant derivation of this result using the Mellin-Perron technique can be found in [8]. An illuminating discussion of the properties of this function is given in [4], as well as an extensive bibliography and discussion of the literature on digital sums. We will not make use of this result, except in the most general sense of referring to the properties of digital sums.

One further result worthy of note is that the ordinary generating function of the digital sum $s_{b}(n)$ is given by Adams-Walter and Ruskejin in [1], and is

[TABLE]

In the next section we prove that the definition of SanD numbers and primes restricts the differences $q-p$ to a given subset of the integers. In Section 3 we study the growth in the number of SanD numbers and primes, and give probabilistic arguments that the number of decimal SanD numbers less than $x$ grows as $(2/3)x$ as $x$ gets large, while the number of decimal SanD primes grows like $(3/4)x/\log^{2}{x}.$ In Section 4 we consider SanD primes with an arbitrary base $b.$ The number of such primes less than $x$ is also expected to grow as $c_{b}x/\log^{2}{x},$ and we calculate the constant $c_{b}.$ We show that $c_{b}=0$ when $b$ is odd. In Section 5 we give numerical results, notably the number of SanD primes less than $3\cdot 10^{12},$ and show that the numerical data gives results consistent with the probabilistic arguments of the earlier section. Section 6 treats the case of binary SanD primes, which are also enumerated up to $3\cdot 10^{12},$ and analysed. The next section gives an heuristic calculation of the number of SanD numbers less than $x$ by approximating the sum-of-digits function $s_{10}(p\cdot q)$ by an appropriately chosen Gaussian random variable. This gives rise to results in qualitative, though not quantitative agreement with the numerical data. We then compare this behaviour to that of the SanD primes.

2 Possible values of $\Delta$ for SanD numbers and SanD primes.

2.1 SanD numbers

Lemma 1**.**

For base-10 SanD numbers, $\Delta\equiv 5$ (mod 9) or $\Delta\equiv 0$ (mod 9)

Proof.

Any natural number $n$ can be written, in decimal form, as

[TABLE]

Its digital sum, $s_{10}(n)=\sum_{k}\alpha_{k}.$ Since $\alpha_{k}\cdot 10^{k}\equiv\alpha_{k},$ (mod 9), working in (mod 9) it follows that every number is equal to the sum of its digits.

For SanD numbers we require that $s_{10}(n(n+\Delta))=\Delta.$ So $n(n+\Delta)-\Delta\equiv 0$ (mod 9) or $(n-1)(n+\Delta+1)\equiv 8$ (mod 9). This excludes the values $n+\Delta\equiv 2,\,5,\,8$ (mod 9). This leaves the values $n+\Delta\equiv 0,\,3\,,6$ (mod 9) and $n+\Delta\equiv 1\,,4\,,7$ (mod 9). In the first case we have $\Delta\equiv 0$ (mod 9) and in the second case $\Delta\equiv 5$ (mod 9). Thus possible values of $\Delta$ are $9k$ and $5+9k,$ $k=1,2,3,4,\ldots.$

Corollary 2**.**

The condition $\Delta\equiv 0$ (mod 9) implies that the SanD numbers $(n,n+\Delta)\equiv(0,0)$ (mod 3).

Proof.

$n(n+\Delta)=n^{2}+\Delta n.$ If $\Delta\equiv 0$ (mod 9), then $\Delta\equiv 0$ (mod 3) and so $n^{2}\equiv 0$ (mod 3), hence $n\equiv 0$ (mod 3).

Corollary 3**.**

The condition $\Delta\equiv 5$ (mod 9) implies that the SanD numbers $(n,n+\Delta)\equiv(2,1)$ (mod 3).

Proof.

$(n,n+\Delta)\equiv 5$ (mod 9) so $n^{2}+5n\equiv 5$ (mod 9), which has solution $n\equiv 2$ (mod 3), hence $n+\Delta\equiv 1$ (mod 3).

2.2 SanD primes

Lemma 4**.**

For base-10 SanD primes, $\Delta\equiv 5$ (mod 9). If $\Delta$ is odd, the only prime-pair is $(2,7).$ If $\Delta$ is even, then $\Delta=14+18k,$ with $k=0,1,2,3,4,\ldots.$

Proof.

For SanD primes we require that $s_{10}(p(p+\Delta))=\Delta.$ So $p(p+\Delta)-\Delta\equiv 0$ (mod 9) or $(p-1)(p+\Delta+1)\equiv 8$ (mod 9). This excludes the values $p+\Delta\equiv 2,\,5,\,8$ (mod 9), and since $p+\Delta$ is prime, the values $p+\Delta\equiv 3,\,6\,,9$ (mod 9) are also excluded. This leaves $p+\Delta\equiv 1\,,4\,,7$ (mod 9) giving $p\equiv 5\,,8\,,2$ respectively. In each case we have $\Delta\equiv 5$ (mod 9). If $\Delta$ is odd, the only solution is $p=2,\,p+\Delta=7,$ as for other primes $p,$ $p+\Delta$ is even. If $\Delta$ is even the only solutions are $\Delta=14+18k,$ with $k=0,1,2,3,4,\ldots.$

Corollary 5**.**

The condition $\Delta\equiv 5$ (mod 9) implies that the SanD prime pair $(p,p+\Delta)\equiv(2,1)$ (mod 3).

Proof.

For the prime pair $(2,7)$ the result is immediate by inspection. Otherwise the proof is identical to that of the preceding corollary.

3 The conjectured asymptotic behaviour of SanD numbers and SanD primes

In this section we give arguments, but not proofs, that the number of SanD numbers less than $x$ grows as $\frac{2}{3}x$ as $x$ gets large, while the corresponding result for SanD primes is $\frac{3x}{4\log^{2}{x}}.$ The absence of proofs is hardly surprising since even without the extra conditions that define SanD primes, no results for prime pairs $(p,q)$ with fixed gap $\Delta=q-p$ have been proved, despite the remarkable recent developments described in the papers of Zhang [18] and Maynard [13].

3.1 SanD numbers

Base-10 SanD numbers less than $x$ are defined as the set of ordered pairs $(a,b)$ such that $1\leq a<b\leq x$ and $b-a=s_{10}(a\cdot b).$

There are $x(x-1)/2\sim x^{2}/2$ choices for the pair $(a,b)$ such that $1\leq a<b\leq x.$ The digital sum constraint implies that $s_{10}(a\cdot b)\equiv 5$ (mod 9) or [math] (mod 9). We conjecture that this constraint reduces the quadratic growth of number pairs to linear growth. To see this, first note that $b-a=s_{10}(a^{2})$ has exactly one solution for each $a,$ namely $b=s_{10}(a^{2})+a.$ So asymptotically there are precisely $x$ such numbers $\leq x.$ However it is not true that $b-a=s_{10}(ab)$ has a solution $b$ for every $a,$ and it is also possible (though it occurs infrequently) that for some values of $a$ there is more than one solution $b.$ Accordingly, we write $c_{1}x$ for the number of SanD numbers less than or equal to $x$ solving $b-a=s_{10}(a\cdot b).$

A totally different, but more complicated argument is the following: In 1968 Kátai and Mogyoródi [10] proved the asymptotic normality of the sum-of-digits function with mean $M=(9/2)\log_{10}{x}$ (this was known since 1940, [2]), and variance $V=(33/4)\log_{10}(x).$ Then $s_{10}(a\cdot b)=b-a$ holds with a probability that is, for each potential pair $(a,b)$ given by the Gaussian

[TABLE]

Since both $M$ and $V$ are very small compared to $x,$ all pairs $(a,b)$ occurring with appreciable probability have $a$ and $b$ close to the square-root of $x.$ Thus $a\cdot b\sim const.x.$

From corollaries 2 and 3, SanD numbers must satisfy

[TABLE]

Since there are nine equally likely values for $(a,\,b)$ (mod 3), this gives a probability of $2/9$ that pairs chosen at random satisfy these conditions. Choosing two numbers at random, their product is equally likely to be 0, 1 or 2 (mod 3), so each product has probability $1/3.$ The ratio of these probabilities, $2/3,$ is the constant $c_{1}$ above, so the number of SanD numbers is expected to grow like $2x/3.$ Numerical experimentation is consistent with this result.

3.2 SanD primes.

The fact that the pair $p,\,\,p+\Delta$ are both primes suggests the (generalized) twin-prime conjecture, albeit constrained by the stringent condition on the digital sum of the product.

As discussed, for example, by Tao in [16], the primes are believed (not proved) to behave pseudo-randomly. This belief goes back at least to Cramér [5], whose model can be easily refined, since all primes greater than 2 are odd, to one in which primes $<x$ are modelled by a set of integers such that odd integers are selected with probability $2/x.$ Further refinement of this model, as discussed in [16] leads to the prediction that the number of twin primes $<x$ behaves as $2C_{2}\frac{x}{\log^{2}{x}},$ where

[TABLE]

and is known as the Hardy-Littlewood constant [9]. Subdominant terms are given by the stronger conjecture that the number of twin primes $<x$ is asymptotically $2C_{2}\operatorname{Li}_{2}(x).$ Considerably greater detail is to be found in [17].

There are known deficiencies in the refined Cramér model, particularly for local problems. Maier [12] obtained the (then) surprising result that the model was defective for certain short intervals between primes, while Pintz [15] showed further problems, of a global nature. Despite this, the refined Cramér model does seem to predict what is believed to be the correct asymptotic behaviour of twin primes, including the Hardy-Littlewood constant [9].

At a similar level of assumption then, the number of unconstrained prime SanD pairs $(p,p+\Delta)<x$ is expected to behave as $c\cdot x/\log^{2}{x},$ where the constant $c$ depends on $\Delta$111The dependence on $\Delta$ is irregular, depending on the prime divisors of $\Delta.$ See for example [3]. Clearly, $c(2)=C_{2},$ as defined above..

In the case of SanD primes, we have shown that $\Delta=14+18k,$ $k=0,1,2,\ldots,$ (neglecting the isolated case $\Delta=5$). However for $x=10^{k},$ the number of possible choices for $\Delta$ increases roughly as $\log_{10}{k}.$ For example, for $x=10^{8}$ there are exactly 8 values of $\Delta$ contributing to the total number of SanD primes $<10^{8},$ as can be seen from Table 1 below. This would imply an extra factor $\log{x}$ in the asymptotic behaviour of SanD primes.

There is however a second constraint, which is that the digital sum must be equal to $\Delta.$ The summands of the digits of the natural numbers up to $10^{n}$ vary from 1 to $9\log_{10}{x},$ that is, from 1 to $9n.$ The distribution is symmetrical and unimodal. Since the number of summands is proportional to $\log{x},$ the probability of a particular summand is proportional to $1/\log{x}.$ Similarly, restricting ourselves to primes, or even twin primes, the number of summands still appears to be proportional to $\log{x},$ so the probability of a particular summand is given by the reciprocal, $1/\log{x}.$

Thus we see that these two effects, the infinite number of possible values for $\Delta$ and the constraint that the digital sum of the product $s_{10}(p\cdot(p+\Delta))=\Delta,$ cancel each other out. So we expect that, asymptotically, the number of SanD primes $<x$ grows as $const\cdot x/\log^{2}{x}.$

Despite the superficial similarity to twin primes discussed above, it is more appropriate to compare the SanD prime pairs with uncorrelated pairs of prime numbers. So we will compare the number $N_{1}$ of prime pairs $(a,b),$ assuming ordering $a<b,$ with $b-a=s_{10}(a\cdot b),$ and $b<x,$ with the total number $N_{2}$ of prime pairs $(a,b)$ in this range.

We are interested in the ratio

[TABLE]

The number $N_{2}$ of uncorrelated pairs is simply the square of the number of primes in this range. The Prime Number Theorem tells us that

[TABLE]

asymptotically for large $x$ where the factor $1/2$ comes from the ordering.

The main statistical assumption is that the ratio $r$ is a product of factors, one for each prime divisor $q,$ with the divisibility of the candidate primes by different divisors $q$ being uncorrelated. For each $q,$ the factor is the ratio of probabilities of integer-pairs being both prime to $q,$ with and without the digit-sum condition.

For every prime $q$ not equal to 3, the digit-sums are distributed randomly over all the residue classes (mod $q$).

For each of these primes, the digit-sum condition does not change the probability that an integer-pair will both be prime to $q.$ Each of these primes contributes a factor unity to the ratio $r.$ Only for $q=3$ does the digit-sum condition change the probabilities.

Since the digit-sum is equal to $(a\cdot b)$ (mod 3), the pair $(a,b)$ must always be $(2,1)$ (mod 3), as proved in corollary 5.

The chance that the elements of an uncorrelated pair $(a,b)$ are both prime to 3 is $(4/9),$ while a pair satisfying the digit-sum condition must be $(2,1)$ (mod 3) or $(0,0)$ (mod 3), as proved in corollaries 2 and 3. Only in the first case are both prime to 3, so the probability is $(1/2).$ The factor contributed by the prime 3 to the ratio $r$ is then

[TABLE]

Multiplying all the factors together gives the result

[TABLE]

where $T_{1}$ and $T_{2}$ are the total number of integer pairs with and without the digit-sum condition respectively. We calculated

[TABLE]

the number of SanD numbers $<x,$ in subsection 3.1, while

[TABLE]

This gives the final result, as $x$ tends to infinity,

[TABLE]

4 SanD primes with an arbitrary base.

Generalising the above result to an arbitrary base, we find that for base-$b$, the number of SanD primes less than $x$ as $x$ tends to infinity, grows as $c_{b}x/\log^{2}{x},$ where

[TABLE]

where the product is taken over prime factors $q$ of $b-1.$ (The similarity of this constant to the Hardy-Littlewood constant is noteworthy).

This result follows from the generalisation of the statistical argument given above for the decimal case, calculating the ratio $r=N_{1}/N_{2}.$ This ratio is, as stated, a product of factors, one for each prime divisor of $q,$ with the divisibility of the candidate primes by different divisors $q$ being uncorrelated.

It follows from Legendre’s result (1) that the digit-sums are randomly distributed over all the residue classes (mod $q$) except for prime factors of $b-1$. (This gave $q=3$ as the only case in the decimal case $b=10$ we originally considered. Now we have the same result for base 4, as $q=3$ is the only prime factor of $b-1=3,$ while for bases 6 and 8 the only prime factors we need consider are 5 and 7 respectively. For base 16 we’d need to consider both 3 and 5).

So the probability that the elements of an uncorrelated pair $(a,b)$ are both prime to $q$ is $((q-1)/q)^{2}.$ We have already seen that, modulo 3, a pair satisfying the digit sum condition must be (2,1) or (0,0). Only in the first case are both prime to 3, so the relevant probability is 1/2. Now generalising this, we see that for mod 5 the relevant pairs are (0,0), (3,1), (4,2), (2,3), and only in the last three cases are both prime to 5, giving a factor 3/4. And in general this factor will be $(q-2)/(q-1).$ Thus

[TABLE]

As before $T_{2}=x^{2}/2$ and $T_{1}=(q-1)/q,$ which follows by generalising the argument in Section 3 as follows: The probability of a randomly chosen pair satisfying the divisibility condition is $(q-1)/q^{2},$ and the probability of a particular product is $1/q$, so this ratio is $(q-1)/q,$ given as 2/3 for the decimal case, where $q=3.$ Putting these factors together gives the result. It follows that $c_{b}=0$ for odd bases $b.$ That is to say, there are no SanD primes in such cases. For $b=2$ one has $c_{2}=1.$

The analogue of Lemma 4 for base 2 SanD primes is: $\Delta=4+2k,\,\,k=0,1,2,\ldots,$ and $c_{2}(base_{2})=1.$

For base 4 SanD primes it is: $\Delta=8+6k,\,\,k=0,1,2,\ldots,$ and $c_{2}(base_{4})=3/4.$

For base 6 SanD primes it is: $\Delta=6+10k;\,\,8+10k,\,\,k=0,1,2,\ldots,$ and $c_{2}(base_{6})=15/16.$

For base 8 SanD primes it is: $\Delta=10+14k;\,\,18+14k;\,\,20+14k,\,\,k=0,1,2,\ldots,$ and $c_{2}(base_{8})=35/36.$

5 Numerical calculation of SanD numbers and primes.

We first wrote a Maple program to enumerate SanD primes. We wanted to provide numerical support for the conjectured behaviour, notably that the number of SanD primes $<x$ grows as $c_{2}x/\log^{2}{x}$ with $c_{2}=3/4.$

In a few hours on a 4GHz Intel i7 iMac with 64Gb of memory we found all SanD primes as large as $3\cdot 10^{8},$ but convergence was irregular. Andrew Conway kindly wrote a C program that, on a larger computer with 32 cores and 256 Gb of memory enabled us to obtain SanD primes as large as $3\cdot 10^{12}$ in a day of computing time.

In Table 1 below we give the number of SanD primes less than $x$ for various values of $x\leq 3\cdot 10^{12},$ given with the appropriate value of $\Delta.$ We have seen that with $\Delta=5$ there is only one SanD prime. With $\Delta=14$ there appears to be only 19. This is misleading. There is a large gap to the next one, which is 11000000000000003, that is, around $10^{16},$ which is beyond our enumerative ability. Indeed, for any valid value of $\Delta$ there are (probabilistically) an infinite number of SanD primes. We now sketch a constructive proof for the case $\Delta=14,$ which can be repeated mutatis mutandis for any other valid value of $\Delta.$

Proof: Assume that the primes behave like independent random variables. Consider the number

[TABLE]

with $r,\,s>0.$ Then

[TABLE]

The digital sum $(5+1+2+2+1+1+2)$ is 14 for every such product, so the number of prime-pairs is, probabilistically speaking, infinite. QED.

Similarly, for $\Delta=32,$ for the same number $S,$ $s_{10}(S(S+32))=32.$ For $\Delta=50,$ the appropriate choice is $S=7+3\cdot 10^{r}+10^{s},$ with $r,\,s>0.$ Then $s_{10}(S(S+50))=50.$ Similar such numbers $S$ can be found for other values of $\Delta$, showing that for every valid $\Delta$ there is an infinite number of SanD numbers, and so, probabilistically speaking, an infinite number of SanD primes.

Referring again to Table 1, Richard Brent (private communication) pointed out (i) that the diagonal above which the entries are zero can be immediately predicted from the fact that $s_{10}(n)<9d$ for $n<10^{d},$ (ii) that the maximal entry in each row occurs approximately halfway to the boundary, and (iii) that the above probabilistic argument can be extended to conjecture the growth of $N_{\Delta}(x),$ the number of SanD primes $x$ with $x<X$ and difference $\Delta.$ In particular, that $N_{14}(X)\gg\log\log{X}.$

[FIGURE:]

Assuming that the number of SanD primes less than $x$ grows as $c_{2}\cdot x/\log^{2}{x}$ as argued above, we have estimated the value of the constant $c_{2}$ in three different ways. Firstly, as the number of primes less than $x,$ denoted as usual by $\pi(x),$ grows as $x/\log{x}$, it follows that $xT(x)/\pi(x)^{2}$ should converge to $c_{2}.$ This estimator is given in the third column of Table 2. Another estimator is $T(x)\cdot\log^{2}{x}/x,$ while if the asymptotics are similar to that of twin primes, $T(x)/\operatorname{Li}_{2}(x)$ would converge more rapidly. Recall that asymptotically

[TABLE]

while [7]

[TABLE]

so these differ only in the last quoted coefficient, and even then by only 20%. These last two estimators are given in columns four and five of Table 2. Both seem to fit the SanD distribution somewhat better than the leading term, $x/\log^{2}{x},$ and the same is true for binary SanD primes, discussed below. This may not persist for larger values of $x$ than we are able to compute.

In no case is convergence regular, unlike the corresponding situation for primes or twin primes. This is not surprising as the SanD primes are likely to have jagged irregularities in their distribution because the digit-sum function has jagged irregularities whenever the first or second digit changes from nine to zero.

The data in Table 2 is totally consistent with a value of $c_{2}\approx 0.75.$ Taking data for $x\geq 10^{6},$ the third column entries average around $c_{2}=0.725,$ the fourth column average is $c_{2}=0.811,$ and the fifth column gives $c_{2}=0.721.$ This variation is indicative of the jagged convergence, and an estimate of $c_{2}\approx 0.75$ seems appropriate, in agreement with our calculation above.

6 Binary SanD primes.

We have also investigated the properties of SanD primes in base 2. The number of such SanD primes $B(x)$ less than $x$ for $x=10^{n},\,\,n=2,3,4,\ldots,12$ and $x=3\cdot 10^{n}$ for $n=9,\ldots,12,$ is given in the second column of Table 3. Note that $B(10)=0.$

As with base-10 SanD primes, we write $B(x)\sim b_{2}\cdot x/\log^{2}{x},$ and estimate the constant $b_{2}$ three different ways. The results are shown in Table 3. We see that convergence is significantly smoother than in the base-10 case, but still not monotonic, due to the jagged irregularities in the digit-sum function.

Nevertheless, a glance at the table entries would suggest a limit of 1 and this is as calculated in Section 4. These numbers show clearly the difference between decimal and binary digit-sums. The decimal sum of $x$ differs from $x$ by a multiple of 9, and this causes the bunching of SanD primes into the groups $\Delta=14,\,32,\,50,$ etc. In the binary case the 9 is replaced by 1, and the divisibility by 1 does not cause any bunching. There is only the divisibility by 2 imposed by the fact that all primes after 2 are odd. So we see that the binary coefficients converge to the value 1 rather than 3/4. For the binary case, there is no special prime that plays the role of 3 in the decimal case, and every SanD integer pair of size $x$ has an equal chance $1/\log^{2}{x}$ of being a prime-pair.

7 Irregular convergence

7.1 SanD numbers

In this section we give an heuristic calculation for the irregular behaviour of decimal SanD numbers, based on the approximation that each sum-of-digits function $s_{10}(a\cdot b)$ can be replaced by a Gaussian random variable, with mean value $M=(9/2)\log_{10}(u)$ and variance $V=(33/4)\log_{10}(u),$ where $u=a\cdot b.$ Here (9/2) is the mean value of a decimal digit, and (33/4) is the mean-square-deviation from the mean, as discussed above eqn. (2).

This approximation is good when $u$ is large and the $\log_{10}(u)$ digits are statistically independent variables. Then the equation $s_{10}(a\cdot b)=b-a$ holds with a probability that is for each potential pair $(a,b)$ equal to the Gaussian eqn. (2).

Since $M$ and $V$ are very small compared with $u,$ all pairs that occur with appreciable probability have $a$ and $b$ both close to the square root of $u.$ The potential SanD numbers $(a,b)$ lie in a narrow strip around the line $a=b.$ To accord with the SanD prime calculation, we restrict the allowed values of $b-a$ to be integers of the form $18j-4$ with $j=1,2,3,\ldots.$ Therefore the population density of SanD numbers is given by the sum

[TABLE]

summed over integer $j.$ The sum is strictly over positive $j,$ but we can extend it to all positive and negative $j$ without significant error, since the terms with negative $j$ are much smaller than unity.

The sum $W(u)$ can be transformed to a rapidly converging sum by using the Poisson Summation formula, giving

[TABLE]

We keep only the three terms of the transformed sum with $j=0,1$ and $-1.$ These give

[TABLE]

with exponent

[TABLE]

The omitted terms with $|j|>1$ are of order $u^{-4a}$ or smaller and are certainly negligible. The equation for $W(u)$ shows that the SanD numbers occur with approximately constant population density 1 as a function of the square-root of $u,$ with a deviation which is a low power of $u$ multiplied by a cosine periodic in $\log_{10}(x)$ with period 4.

Since the digit-sums are not in fact independent random variables, this calculation using Gaussian probabilities is not rigorous.

In our previous calculations, we have been counting SanD numbers and primes $(a,b)$ such that $a<b<x,$ and calculating the number of such numbers/primes $<x.$ In the above treatment, we start with the probability $P(u)$ that an integer $u$ is the product $a\cdot b$ of a SanD number pair, so typically $x=\sqrt{u}.$

To test this approximate treatment, we have counted SanD numbers such that $a\cdot b<u,$ for $u=10^{n/5},$ where $n=1,2,\ldots,40.$ Denote these counts $d(n).$ For $n<13,$ $d(n)=0.$ For $n\geq 13$ the counts $d(n)$ are

[TABLE]

for $n=13,14,15,\ldots 40,$ respectively.

The probability $P(u)$ above is then predicted to be

[TABLE]

(The prefactor $1/(12\sqrt{u})$ is included to give the predicted asymptotic behaviour $x/6$ for the number of SanD numbers less than $x.$ The constant $1/12$ arises as we are only counting the subset of SanD numbers corresponding to $b-a=18j-4.$)

To connect the counts $d(n)$ with this formula we require the discrete derivative of the counting function. Thus we define

[TABLE]

To study the fluctuations, we need to compare the calculated value based on the Gaussian approximation

[TABLE]

with the numerical estimate obtained from the data,

[TABLE]

We multiply both $P_{fluc}(u)$ and $D_{fluc}(u)$ by $12\sqrt{10^{n/5}},$ which makes all the fluctuations the same relative scale, and show the results in Figure 1.

The discrete derivatives are shown as (red) circles, the predicted probabilities as (blue) diamonds. The agreement quantitatively is disappointing, but qualitatively is instructive, showing that the actual data and the predicted data display similar irregularities. Unfortunately they don’t correspond in magnitude and phase, presumably because the assumption, that the digit-sums are independent Gaussian random variables, is wrong.

7.2 SanD primes

As discussed above, we expect the number of base-10 SanD primes $<x$ to behave as $(3/4)x/\log^{2}{x}$ as $x$ becomes large. To clearly see the irregular nature of convergence of the numerical data to this behaviour, we compute the deviation as follows. Recall that $T(x)$ denotes the number of SanD primes $<x.$ So

[TABLE]

where $\theta(x)$ is of course unknown. We have calculated $\theta(x)$ for $x<3\cdot 10^{12}$ from the data in Table 2, and show the results in Figure 2. While there is insufficient data to be conclusive, there appears to be similar periodic behaviour to that observed in the SanD number fluctuations above, suggesting a possible periodic correction term.

8 Conclusion

We have defined SanD numbers as ordered pairs $(m,\,n)$ such that the digital-sum $s_{b}(m\cdot n)=n-m=\Delta>0.$ We considered in detail both the decimal ($b=10$) and the binary ($b=2$) case. If both $m$ and $n$ are prime numbers, we refer to SanD primes. Subject to the unproven assumption that primes behave as pseudorandom numbers, in a manner described above, we show that the number of (decimal-based) SanD numbers less than $x$ grows as $c_{1}\cdot x,$ where $c_{1}=2/3,$ while the number of SanD primes less than $x$ grows as $c_{2}\cdot x/\log^{2}{x},$ where $c_{2}=3/4.$ The value of the corresponding constants $c_{b}$ for arbitrary base-$b$ were also calculated. For binary SanD primes we show similarly that the number of such primes $B(x)<x$ behaves as $B(x)\sim b_{2}\cdot x/\log^{2}{x}$ with $b_{2}=1.$

We calculated the number of SanD numbers and primes $<3\cdot 10^{12}$ in order to test the above calculations. The numerical data was consistent with the conjectured results. However due to the sawtooth nature of the digital-sum function, convergence of the estimators of the constants $c_{1}$ and $c_{2}$ with increasing $x$ was found to be more erratic than the corresponding situation with twin primes, which, apart from the constant, have the same leading asymptotics.

The twin prime distribution fits well the SanD prime pair numbers in both the decimal and binary cases (at least for primes less than $3\cdot 10^{12}$), i.e $const\cdot\operatorname{Li}_{2}(x)$ where $const=3/4$ and 1 respectively, in contrast with the twin primes conjecture [9] with $const=2\cdot C_{2}=1.32....$ where $C_{2}$ is the twin prime constant.

9 Acknowledgements

AJG would like to thank Andrew Conway for writing a C program to count SanD primes, Andrew Elvey Price for helpful discussions, Richard Brent for a thorough reading of an earlier version of this paper and many suggested improvements, and Jeffrey Shallit for useful suggestions. He also gratefully acknowledges support from ACEMS the ARC Centre of Excellence for Mathematical and Statistical Frontiers.

10 Appendix

In the table below we show some SanD prime enumerations, giving the first 19 SanD primes for the first few values of $\Delta.$ For each entry $p$ it follows that $s_{10}(p(p+\Delta))=\Delta.$ There is one further entry, not shown, corresponding to the sole SanD prime when $\Delta=5,$ which is $p=2.$