Nov\'ak-Carmichael numbers and shifted primes without large prime factors

TL;DR

This paper establishes new lower bounds for counting Novák-Carmichael numbers, linking their distribution to shifted primes without large prime factors, and provides both unconditional and conditional estimates.

Contribution

It introduces novel lower bounds for Novák-Carmichael numbers' counting function based on properties of shifted primes without large prime factors.

Findings

Unconditional lower bound: x^{0.7039-o(1)}

Conditional lower bound involving exponential decay

Results connect Novák-Carmichael numbers to shifted primes without large prime factors

Abstract

We prove some new lower bounds for the counting function $\mathcal N_{\mathcal C}(x)$ of the set of Nov\'ak-Carmichael numbers. Our estimates depend on the bounds for the number of shifted primes without large prime factors. In particular, we prove that $\mathcal N_{\mathcal C}(x) \gg x^{0.7039-o(1)}$ unconditionally and that $\mathcal N_{\mathcal C}(x) \gg xe^{-(7+o(1))(\log x)\frac{\log\log\log x}{\log\log x}}$, under some reasonable hypothesis.

Equations

$$\Pi (z,z^u)=z^{1+o(1)},$$

$${\mathcal{N}}_{\mathcal{C}}(x)\gg x^{1-u+o(1)}$$

$${\mathcal{N}}_{\mathcal{C}}(x)\gg x^{1-\beta +o(1)},$$

$$\Pi (z,z^u){\asymp }_u\pi (z)\sim \frac{z}{\log z}.$$

$$\frac{\Pi (z,y(z))}{\pi (z)}\sim \frac{\Psi (z,y(z))}{z}$$

$$\Psi (z,e^{\sqrt{\log z}})=z{e}^{-(1/2+o(1))\sqrt{\log z}\log \log z}$$

$$\Pi (z,e^{\sqrt{\log z}})\gg z{e}^{-(1/2+o(1))\sqrt{\log z}\log \log z}.$$

$$\Pi (z,e^{\sqrt{\log z}})\gg z{e}^{-c\sqrt{\log x}\log \log x}$$

$${\mathcal{N}}_{\mathcal{C}}(x)\gg x{e}^{-d\log x\frac{\log \log \log x}{\log \log x}}.$$

$${\mathcal{N}}_{\mathcal{C}}(x)\gg x{e}^{-(7+o(1))\log x\frac{\log \log \log x}{\log \log x}}.$$

$$a^n\equiv 1(\mathrm{mod}{p}_k^{{\alpha }_k}).$$

$$a\equiv g_k(\mathrm{mod}{p}_k^{{\alpha }_k})$$

$$g_k^n\equiv a^n\equiv 1(\mathrm{mod}{p}_k^{{\alpha }_k}).$$

$$a^n=(a^{\phi (p_k^{{\alpha }_k})}{)}^{n/\phi (p_k^{{\alpha }_k})}\equiv 1(\mathrm{mod}{p}_k^{{\alpha }_k})$$

$$a^n=(a^{\phi (2^{\alpha })}{)}^{n/\phi (2^{\alpha })}\equiv 1(\mathrm{mod}{2}^{\alpha }).$$

$$a^n\equiv 1(\mathrm{mod}n),$$

$$\left(\genfrac{}{}{0ex}{}{a}{b}\right)⩾{\left(\frac{a}{b}\right)}^b.$$

$$\left(\genfrac{}{}{0ex}{}{a}{b}\right)=a={\left(\frac{a}{1}\right)}^1.$$

$$\left(\genfrac{}{}{0ex}{}{a}{c+1}\right)=\left(\genfrac{}{}{0ex}{}{a}{c}\right)\frac{a-c}{c+1}⩾{\left(\frac{a}{c}\right)}^c\frac{a-c}{c+1}={\left(\frac{a}{c+1}\right)}^{c+1}\left(1-\frac{c}{a}\right){\left(1+\frac{1}{c}\right)}^c.$$

$$\left(\genfrac{}{}{0ex}{}{a}{c+1}\right)⩾{\left(\frac{a}{c+1}\right)}^{c+1}$$

$$\mathcal{D}(s,r)=\prod _{p⩽r}{p}^{\left[\frac{\log s}{\log p}\right]},$$

$$\log \mathcal{D}(s,r)=O\left(\frac{r\log s}{\log r}\right)$$

$$\mathcal{E}(\mathcal{A},s,r)=\mathcal{D}(s,r)\prod _{p\in \mathcal{A}}p$$

$$\log \mathcal{D}(s,r)=\sum _{p⩽r}\left[\frac{\log s}{\log p}\right]\log p⩽\sum _{p⩽r}(\log p)\frac{\log s}{\log p}=\pi (r)\log s=O\left(\frac{r\log s}{\log r}\right).$$

$$r=\frac{\log x}{\log {\log }^{2}x},s=r^{1/u}$$

$$A=\left[u\frac{\log x}{\log r}-u\frac{\log \mathcal{D}(s,r)}{\log r}\right].$$

$$\log \mathcal{D}(s,r)=O\left(\frac{r\log s}{\log r}\right)=O\left(\frac{\log x}{\log \log x}\right)=o(\log x)$$

$$\mathcal{E}(\mathcal{A},s,r)=\mathcal{D}(s,r)\prod _{p\in \mathcal{A}}p⩽\mathcal{D}(s,r)s^A=e^{\log \mathcal{D}(s,r)+A\log s}.$$

$$\log \mathcal{E}(\mathcal{A},s,r)⩽\log \mathcal{D}(s,r)+A\log s⩽\log x.$$

$$\mathcal{E}(\mathcal{A},s,r)=\mathcal{D}(s,r)\prod _{p\in \mathcal{A}}p=\mathcal{D}(s,r)\prod _{p\in \mathcal{B}}p=\mathcal{E}(\mathcal{B},s,r)$$

Full text

00footnotetext: The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. № 14.641.31.0001, the Simons Foundation and the Moebius Contest Foundation for Young Scientists

1 Introduction

In the paper [7], author introduced the Novák-Carmichael numbers. Positive integer $N$ is called a Novák-Carmichael number if for any $a$ coprime to $N$ the congruence $a^{N}\equiv 1\pmod{N}$ holds. Later, S. V. Konyagin posed a problem about the order of growth of the quantity $\mathcal{N}_{\mathcal{C}}(x)$ — the number of Novák-Carmichael numbers which are less than or equal to $x$. The present work provides a partial answer to this question.

It turns out that the lower bounds for the quantity $\mathcal{N}_{\mathcal{C}}(x)$ can be deduced from the theorems on the distribution of shifted prime numbers without large prime factors. Namely, for a positive integers $x$ and $y$ denote by $\mathcal{P}(x,y)$ the set of all prime numbers $p\leqslant x$ such that the largest prime factor of $p-1$ is less than or equal to $y$. Let also $\Pi(x,y)$ be the number of elements of the set $\mathcal{P}(x,y)$. Then the following proposition holds:

Theorem 1

Let $u$ be some fixed real number with $0<u<1$. If for $z\to+\infty$ we have

[TABLE]

then the lower bound

[TABLE]

holds.

Lower bounds for the quantity $\Pi(z,z^{u})$ for different values of $u$ are studied in the papers [9],[5],[3]. In particular, using the result of the last article we obtain

Corollary 1

The inequality

[TABLE]

is true for $\beta=0.2961$.

Remark 1

It is conjectured that for any fixed positive $u$ we have

[TABLE]

It is also reasonable to conjecture that for any nice enough function $y(z)$ the asymptotic relation

[TABLE]

holds, where $\Psi(z,y)$ is the number of natural numbers $n\leqslant z$ such that the largest prime factor of $n$ is less than or equal to $y$.

For example, if we assume that relation (1.1) is true for $y(z)=e^{\sqrt{\log z}}$, then by the formula

[TABLE]

(see [6]) we get

[TABLE]

Using a slightly weaker form of this assumption, we improve the estimate of the Theorem 1:

Theorem 2

Suppose that for some fixed constant $c>0$ the inequality

[TABLE]

holds. Then for any $d>6+2c$ we have

[TABLE]

In particular, if relation (1.1) is true for $y(z)=e^{\sqrt{\log z}}$, then

[TABLE]

2 Proofs of the theorems

The constructions that we will use in our proofs are largely similar to that of papers [9], [1]. First of all, we need a description of Novák-Carmichael numbers in terms of their prime factors, which is an analogue of Koselt’s criterion (cf. [8]) for Carmichael numbers:

Lemma 1

Natural number $n$ is a Novák-Carmichael number if and only if for any prime divisor $p$ of $n$ the number $p-1$ also divides $n$.

Proof

Let $n=2^{\alpha}\prod_{k=1}^{m}p_{k}^{\alpha_{k}}$, where $p_{k}$ are distinct odd prime numbers and $\alpha_{k}\geqslant 1$. If $n$ is a Novák-Carmichael number, then for any $a$ coprime to $n$ and any $k$ we have

[TABLE]

On the other hand, by the Chinese remainder theorem we can choose $a$ such that for any $k$ the congruence

[TABLE]

holds, where $g_{k}$ is some primitive root modulo $p_{k}^{\alpha_{k}}$.

Consequently, for any $k$ we have

[TABLE]

Thus, for any $k$ the number $n$ is divisible by the multiplicative order of $g_{k}$ modulo $p_{k}^{\alpha_{k}}$. Hence $p_{k}^{\alpha_{k}-1}(p_{k}-1)$ divides $n$. So, for any odd prime divisor $p$ of $n$ we have $p-1\mid n$. Also, $2-1$ divides $n$.

Conversely, if for any prime $p$ dividing $n$ the number $p-1$ also divides $n$, then for any $k$ we have $\varphi(p_{k}^{\alpha_{k}})\mid n$ and $\varphi(2^{\alpha})\mid n$. Hence, if $(a,n)=1$, then

[TABLE]

for any $k$ and

[TABLE]

From these congruences and pairwise coprimality of numbers $2^{\alpha},p_{1}^{\alpha_{1}},\ldots,p_{m}^{\alpha_{m}}$ we obtain

[TABLE]

as needed.

For the asymptotic estimates of sizes of certain sets the following inequality involving binomial coefficients is needed:

Lemma 2

Let $a$ and $b$ be a positive integers with $b\leqslant a/2+1$. Then we have

[TABLE]

Proof

Let us prove this statement by induction over $b$.

The case $b=1$ is obvious, since

[TABLE]

Suppose now that $1<c+1\leqslant a/2+1$ and the inequality is true for $b=c$. Then we have

[TABLE]

On the other hand, $c\leqslant a/2$ and $(1+\frac{1}{c})^{c}\geqslant 2$, so the inequality

[TABLE]

holds, which was to be proved.

In the next lemma, for arbitrary real numbers $r$ and $s$ satisfying the inequality $2\leqslant r\leqslant s$ we will construct the number $\mathcal{D}(r,s)$ with some remarkable properties.

Lemma 3

Let $s,r\in\mathbb{R}$ and $2\leqslant r\leqslant s$. If

[TABLE]

where the product is taken over prime numbers $p$, then

[TABLE]

and for any subset $\mathcal{A}\subseteq\mathcal{P}(s,r)$ the number

[TABLE]

is a Novák-Carmichael number.

Proof

Indeed,

[TABLE]

Let us prove now that the number $\mathcal{E}(\mathcal{A},s,r)$ is a Novák-Carmichael number. Suppose that $q$ is a prime factor of $\mathcal{E}(\mathcal{A},s,r)$. Then we have either $q\mid\mathcal{D}(s,r)$ or $q\in\mathcal{P}(s,r)$. But all the prime factors of $\mathcal{D}(s,r)$ are not exceeding $r$ and so are lying in $\mathcal{P}(s,r)$. Thus, $q\in\mathcal{P}(s,r)$.

Consequently, $q-1=\prod_{l=1}^{t}p_{l}^{\beta_{l}}$ and $p_{l}\leqslant r$ for any $l$. On the other hand, $p_{l}^{\beta_{l}}\leqslant q-1<s$. Taking the logarithms, we obtain $\beta_{l}\leqslant\left[\frac{\log s}{\log p_{l}}\right]$. Thus, for any $l$ we have $p_{l}^{\beta_{l}}\mid\mathcal{D}(s,r)$, so $q-1\mid\mathcal{D}(s,r)\mid\mathcal{E}(\mathcal{A},s,r)$. Hence, by the Lemma 1, our number is a Novák-Carmichael number. This concludes the proof.

Let us now prove Theorems 1 and 2.

Proof of Theorem 1

Suppose that $0<u<1$ and $\Pi(z,z^{u})=z^{1+o(1)}$ as $z\to\infty$. We introduce the notation

[TABLE]

and

[TABLE]

By the Lemma 3 we have

[TABLE]

hence, $A=(u+o(1))\frac{\log x}{\log r}$. Now, for any subset $\mathcal{A}\subseteq\mathcal{P}(s,r)$ of cardinality $A$ consider the number $\mathcal{E}(\mathcal{A},s,r)$. By the Lemma 3 this number is a Novák-Carmichael number and

[TABLE]

Note that $A=\left[u\frac{\log x}{\log r}-u\frac{\log\mathcal{D}(s,r)}{\log r}\right]\leqslant u\frac{\log x}{\log r}-u\frac{\log\mathcal{D}(s,r)}{\log r}=\frac{\log x}{\log s}-\frac{\log\mathcal{D}(s,r)}{\log s}$. From this we obtain the inequality

[TABLE]

Hence, all the constructed numbers $\mathcal{E}(\mathcal{A},s,r)$ are less than or equal to $x$. Furthermore, all these numbers are distinct, as otherwise for some different subsets $\mathcal{A},\mathcal{B}$ we would have had

[TABLE]

hence, $\prod\limits_{p\in\mathcal{A}}p=\prod\limits_{p\in\mathcal{B}}p$, which is not the case.

So, the number of Novák-Carmichael numbers not exceeding $x$ is at least as large as the number of subsets in $\mathcal{P}(s,r)$ of cardinality $A$. But for large enough $x$ we have

[TABLE]

Consequently, using Lemma 2 we get

[TABLE]

From

[TABLE]

and

[TABLE]

we finally get

[TABLE]

which is the required result.

The proof of Theorem 2 is proceeded analogously. All we need is some different choice of parameters $r,s$ and $A$.

Proof of Theorem 2

Assume that $\Pi(z,e^{\sqrt{\log z}})\gg ze^{-c\sqrt{\log z}\log\log z}$. Let us choose

[TABLE]

and, as before,

[TABLE]

Now, similarly to the proof of Theorem 1, considering the subsets of $\mathcal{P}(s,r)$ which contain exactly $A$ elements we obtain

[TABLE]

Furthermore, by Lemma 3 we have $A=\frac{\log x}{\log s}+O\left(\frac{r\log s}{\log r}\right)\geqslant\frac{\log x}{(\log\log x)^{2}}$. Also, due to the assumption of the theorem, we have $\Pi(s,r)\gg se^{-c\sqrt{\log s}\log\log s}\geqslant se^{-2c\log x\log\log x}$. So, for any $d>6+2c$ the inequality

[TABLE]

holds.

Thus, we have

[TABLE]

which concludes the proof of Theorem 2.

3 Conclusion

We showed that lower bounds for the number of shifted prime numbers without large prime factors imply some nice lower bounds for the counting function of the set of Novák-Carmichael numbers. It is a well-known fact that these theorems also provide estimates for the counting function of Carmichael numbers (cf. [2]). However, in our situation it is possible to use much simplier constructions.

Furthermore, the relation (1.1) for $y(z)=e^{\sqrt{\log z}}$ implies the lower bound which is as strong as the upper bound for the number of Carmichael numbers less than a given magnitude proved by P. Erdös. Unfortunately, the methods of the paper [4] do not allow a direct generalization to the case of Novák-Carmichael numbers. So, the problem of obtaining the correct order of growth of the quantity $\mathcal{N}_{\mathcal{C}}(x)$ remains open even on the assumption of the relation (1.1).