Carl Pomerance

Dartmouth College

This corrigendum fixes a number of small errors/omissions in [J. Eur. Math. Soc. 23, 667–700 (2021)], which in particular affect the numerical values of the exponents of log log x in Theorem 1 and its corollaries.

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum ∑1/(n log n) for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjectu...

We study the asymptotic density of the set of subscripts of the Bernoulli numbers having a given denominator. We also study the distribution of distinct Bernoulli denominators and some related problems.

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty...

We characterize which residue classes contain infinitely many totients (values of Euler's function) and which do not. We show that the union of all residue classes that are totient-free has asymptotic density 3/4, that is, almost all numbers that are 2 mod 4 are in a residue class that is totient-free. In the other direction, we show the existence...

A set of integers greater than 1 is primitive if no element divides another. Erd\H{o}s proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. We answer the Erd\H{o}s question in the affir...

The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let pr$(n)$ denote the largest prime $p$ with $p\# \mid \phi(n)$, where $\phi$ is Euler's totient function. We show that the normal order of pr$(n)$ is $\log\log n/\log\log\log n$. That is, pr$(n) \sim \log\log n/\log\log\log n$ as $n\to\infty$ on a set of integers of asympto...

In an earlier paper we considered the distribution of integers $n$ for which Euler’s totient function at $n$ has all small prime factors. Here we obtain an improvement that is likely to be best possible.

A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an upper bound on the number of primes up to x that belong to a symmetric pair. In the present paper, that theorem is improved to what is likely to be the best possible resu...

A subset of the integers larger than 1 is primitive if no member divides another. Erdős proved in 1935 that the sum of 1 / ( a log ⁡ a ) 1/(a\log a) for a a running over a primitive set A A is universally bounded over all choices for A A . In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some...

Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the nonreal zeros of ζ ( s ) \zeta (s) , that the set of real numbers x ≥ 2 x\ge 2 for which π ( x ) > li ⁡ ( x ) \pi (x)>\operatorname {li}(x) has a logarithmic density, which they computed to be about 2.6 × 10 − 7 2.6\times...

The study of the relative size of the prime counting function $\pi(x)$ and the logarithmic integral li$(x)$ has led to a wealth of results over the past century. One such result, due to Rubinstein and Sarnak and conditional on the Riemann hypothesis (RH) and a linear independence hypothesis (LI) on the imaginary parts of the zeros of $\zeta(s)$, is...

We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $\varphi(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.

A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on sev...

To each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of at most $C_0$ residue classes modulo $p$, whose cardinality $|I_p|$ is equal to 1 on the average. We show that for sufficiently large $x$, the sifted set $\{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\}$ contains gaps of size $ x (\log x)^{1/\...

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that for sufficiently large $x$, the sifted set $\{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\}$ contains gaps of size at least $x (...

The sequence $3, 5, 9, 11, 15, 19, 21, 25, 29, 35,\dots$ consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the Erd\H...

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d<n}d$. Erd\H{o}s-Granville-Pomerance-Spiro conjectured that for any set $\mathcal{A}$ of asymptotic density zero, the preimage set $s^{-1}(\mathcal{A})$ also has density zero. We prove a weak form of this conjecture: If $\epsilon(x)$ is any function tending...

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d<n}d$. Erd\H{o}s-Granville-Pomerance-Spiro conjectured that for any set $\mathcal{A}$ of asymptotic density zero, the preimage set $s^{-1}(\mathcal{A})$ also has density zero. We prove a weak form of this conjecture: If $\epsilon(x)$ is any function tending...

There is a large literature on the asymptotic distribution of numbers free of large prime factors, so-called $\textit{smooth}$ or $\textit{friable}$ numbers. But there is very little known about this distribution that is numerically explicit. In this paper we follow the general plan for the saddle point argument of Hildebrand and Tenenbaum, giving...

There is a large literature on the asymptotic distribution of numbers free of large prime factors, so-called $\textit{smooth}$ or $\textit{friable}$ numbers. But there is very little known about this distribution that is numerically explicit. In this paper we follow the general plan for the saddle point argument of Hildebrand and Tenenbaum, giving...

The sequence $3, 5, 9, 11, 15, 19, 21, 25, 29, 35,\dots$ consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the Erd\H...

Consider the power pseudorandom-number generator in a finite field ${\mathbb F}_q$. That is, for some integer $e\ge2$, one considers the sequence $u,u^e,u^{e^2},\dots$ in ${\mathbb F}_q$ for a given seed $u\in {\mathbb F}_q^\times$. This sequence is eventually periodic. One can consider the number of cycles that exist as the seed $u$ varies over ${...

We investigate the probability that a random odd composite number passes a random Fermat primality test, improving on earlier estimates in moderate ranges. For example, with random numbers to $2^{200}$, our results improve on prior estimates by close to 3 orders of magnitude.

We investigate the probability that a random odd composite number passes a random Fermat primality test, improving on earlier estimates in moderate ranges. For example, with random numbers to $2^{200}$, our results improve on prior estimates by close to 3 orders of magnitude.

We sharpen a 1980 theorem of Erdős and Wagstaff on the distribution of positive integers having a large shifted prime divisor. Specifically, we obtain precise estimates for the quantity $N(x,y):= \#\{n \le x: \ell-1\mid n\text for some \ell-1 > y$, $\ell$ prime $, in essentially the full range of $x$ and $y$. We then present an application to a pro...

We show that for each prime p > 7, every residue mod p can be represented by a squarefree number with largest prime factor at most p. We give two applications to recursive prime generators akin to the one Euclid used to prove the infinitude of primes.

Let σ ( n ) \sigma (n) denote the sum of all of the positive divisors of n n , and let s ( n ) = σ ( n ) − n s(n) = \sigma (n)-n denote the sum of the proper divisors of n n . The functions σ ( ⋅ ) \sigma (\cdot ) and s ( ⋅ ) s(\cdot ) were favorite subjects of investigation by the late Paul Erdős. Here we revisit three themes from Erdős’s work on...

Let sigma be the usual sum-of-divisors function. We say that a and b form a harmonious pair if a/sigma(a) + b/sigma(b) = 1; equivalently, the harmonic mean of sigma(a)/a and sigma(b)/b is 2. For example, 4 and 12 form a harmonious pair, since 4/sigma(4) = 4/7 and 12/sigma(12) = 3/7. Every amicable pair is harmonious, but there are many others. We s...

We establish several related results on Carmichael, Sierpiński and Riesel numbers. First, we prove that almost all odd natural numbers k have the property that 2nk + 1 is not a Carmichael number for any n ∈ N; this implies the existence of a set K of positive lower density such that for any k ∈ K the number 2nk + 1 is neither prime nor Carmichael f...

For a positive integer j we look at the parity of the number of divisors of n that are at most j, proving that for large j, the count is even for most values of n.

We show that the counting function of the set of values of Carmichael's λ - function is x/(log x)η +o(1), where η=1- (1+log log 2)(log 2)= 0:08607.

It is well-known that there are infinitely many irregular primes. We prove a quantitative version of this statement, namely the number of such primes p ≤ x is at least (1 + o(1)) log log x/ log log log x as x → ∞. We show that the same conclusion holds for the irregular primes corresponding to the Euler numbers. Under some conditional results from...

We show that the counting function of the set of values of the Carmichael $\lambda$-function is $x/(\log x)^{\eta+o(1)}$, where $\eta=1-(1+\log\log 2)/(\log 2)=0.08607\ldots$.

Let z(N) be the order of appearance of N in the Fibonacci sequence. This is the smallest positive integer k such that N divides the kth Fibonacci number. We show that each of the six total possible orderings among z(N), z(N + 1), z(N + 2) appears infinitely often. We also show that for each nonzero even integer c and many odd integers c the equatio...

We show that almost all squares are missing from the range of Euler's φ-function.

Letdenote Carmichael's function, so �(n) is the universal ex- ponent for the multiplicative group modulo n. It is closely related to Euler's '-function, but we show here that the image ofis much denser than the image of '. In particular the number of �-values to x exceeds x/(logx) .36 for all large x, while for ' it is equal to x/(logx) 1+o(1) , an...

We study subsets of [1,x] on which the Euler φ-function is monotone (nondecreasing or nonincreasing). For example, we show that for any ϵ>0, every such subset has size smaller than ϵx, once x>x 0(ϵ). This confirms a conjecture of the second author.

It might be argued that elementary number theory began with Pythagoras who noted two-and-a-half millennia ago that 220 and 284 form an amicable pair. That is, if s(n) denotes the sum of the proper divisors of n (“proper divisor” means d │ n and 1 ≤ d < n), then $$s(220) = 284\quad and\quad s(284) = 220.$$When faced with remarkable examples such as...

Let s′(n) = Σd|n, 1<d<nd be the sum of the nontrivial divisors of the natural number n, where nontrivial excludes both 1 and n. For example, s′(20) = 2+4+5+10 = 21. A natural number n is called quasiperfect if s′(n) = n, while n and m are said to form a quasiamicable pair if s′(n) = m and s′(m) = n; in the latter case, both n and m are called quasi...

We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples. Call a natural number as We conclude by discussing some related problems posed by Harborth and Cohen.

We study the distribution of solutions n to the congruence σ(n) ≡ a (mod n). After excluding obvious families of solutions, we show that the number of these n ≤ x is at most x½+o(1), as x → ∞, uniformly for integers a with ∣a∣ ≤ x¼. As a concrete example, the number of composite solutions n ≤ x to the congruence σ(n) ≡ 1 (mod n) is at most x½+o(1)....

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standar...

We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.

This paper studies the maximal size of product-free sets in Z/nZ. These are sets of residues for which there is no solution to ab == c (mod n) with a,b,c in the set. In a previous paper we constructed an infinite sequence of integers (n_i)_{i > 0} and product-free sets S_i in Z/n_iZ such that the density |S_i|/n_i tends to 1 as i tends to infinity,...

For g,n coprime integers, let l_g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l_g(n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichae...

We show that there are sets of integers with asymptotic density arbitrarily close to 1 in which there is no solution to the equation ab=c, with a,b,c in the set. We also consider some natural generalizations, as well as a specific numerical example of a product-free set of integers with asymptotic density greater than 1/2.

Let φ denote Euler's function. Clearly φ(n) | n - 1 if n = 1 or if n is a prime. In 1932, Lehmer asked if any composite numbers n have this property. Improving on some earlier results, we show that the number of composite integers n ≤ x with φ(n) | n - 1 is at most x 1/2/(logx) 1/2+o(1) as x → ∞. Key to the proof are some uniform estimates of the d...

A Fibonacci integer is an integer in the multiplicative group generated by the Fibonacci numbers. For example, 77=21⋅55/(3⋅5) is a Fibonacci integer. Using some results about the structure of this multiplicative group, we determine a near-asymptotic formula for the counting function of the Fibonacci integers, showing that up to x the number of them...

We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler's totient function and σ is the sum-of-divisors function. This proves a 50-year old conjecture of Erd˝ os. Moreover, we show that for some c > 0, there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions. The proofs rel...

Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.

A set of positive integers is said to be primitive if no element of the set is a multiple of another. If $S$ is a primitive set and $S(x)$ is the number of elements of $S$ not exceeding $x$, then a result of Erd\H os implies that $\int_2^\infty (S(t)/t^2\log t) dt$ converges. We establish an approximate converse to this theorem, showing that if $F$...

In this paper we investigate the asymptotic effective- ness of the Gaudry-Hess-Smart Weil descent attack and its gener- alization on the discrete logarithm problem for elliptic curves over characteristic-two finite fields. In particular we obtain nontrivial lower and upper bounds on the smallest possible genus to which it can lead.

We establish a conjecture of Brizolis that for every prime p > 3 there is a primitive root g and an integer x in the interval [1,p − 1] with logg x = x. Here, logg is the discrete logarithm function to the base g for the cyclic group (ℤ/pℤ)×. Tools include a numerically explicit “smoothed” version of the Pólya–Vinogradov inequality for the sum of v...

We obtain the first known power-saving remainder terms for the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields and the mean number of $3$ -torsion elements in the class groups of quadratic fields. In addition, we prove analogous error terms for the density of discriminants of quartic fields and the mean number of...

We let '(¢) and ¿(¢) denote the Euler function and the number-of- divisors function, respectively. In this paper, we study the average value of ¿('(n)) when n ranges in the interval (1;x).

In this note, we look at the radical (that is, the squarefree kernel) of perfect numbers. We raise the question of whether large perfect numbers have the tendency to become far apart from each other and prove several results towards this under the ABC conjecture.

For a natural number n, let λ(n) denote the order of the largest cyclic subgroup of (/n)*. For a given integer a, let Na(x) denote the number of n ≤ x coprime to a for which a has order λ(n) in (/n)*. Let R(n) denote the number of elements of (/n)* with order λ(n). It is natural to compare Na(x) with ∑n≤x R(n)/n. In this paper we show that the aver...

We dedicate this paper to our friend Alf van der Poorten Abstract Assuming a weak version of a conjecture of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m.

For a positive integer n, define s(n) as the sum of the proper divisors of n. If s(n)>0, define s2(n)=s(s(n)), and so on for higher iterates. Sociable numbers are those n with sk(n)=n for some k, the least such k being the order of n. Such numbers have been of interest since antiquity, when order-1 sociables (perfect numbers) and order-2 sociables...

We show that the average and typical ranks in a certain parametric family of elliptic curves described by D. Ulmer tend to infinity as the parameter $d \to\infty$. This is perhaps unexpected since by a result of A. Brumer, the average rank for all elliptic curves over a function field of positive characteristic is asymptotically bounded above by 2....

For a positive integer k letk be the k-fold composition of the Euler function `. In this paper, we study the size of the set fk(n) • xg as x tends to inflnity.

For a real x > 1 and an integer g = 0, ±1, an x-pseudopower to base g is a positive integer which is not a power of g yet is so modulo p for all primes p ≤ x. Let q g (x) denote the least such number. Improving a series of previous results we show that q g (x) ≤ exp(0.86092x) for sufficiently large x. The method is based on a combination of some bo...

We establish a numerically explicit version of the Pólya– Vinogradov inequality for the sum of values of a Dirichlet character on an interval. While the technique of proof is essentially that of Landau from 1918, the result we obtain has better constants than in other numerically explicit versions that have been found more re-cently.

Giuga has conjectured that if the sum of the $(n-1)$-st powers of the residues modulo $n$ is $-1pmod n$, then $n$ is 1 or prime. It is known that any counterexample is a Carmichael number. Lehmer has asked if $varphi(n)$ divides $n-1$, with $varphi$ being Euler&apos;s function, must it be true that $n$ is 1 or prime. No examples are known, but a co...

Introduced by Kraitchik and Lehmer, an $x$-pseudosquare is a positive integer $n\equiv1\pmod 8$ that is a quadratic residue for each odd prime $p\le x$, yet is not a square. We use bounds of character sums to prove that pseudosquares are equidistributed in fairly short intervals. An $x$-pseudopower to base $g$ is a positive integer which is not a p...

An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\le x$. We improve an upper bound for the least such number due to E. Bach, R. Lukes, J. Shallit, and H. C. Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behavi...

We give an introduction to the discrete logarithm problem in cyclic groups and treat the most important methods for solving them. These include the index calculus method, the rho and lambda methods, and the baby steps, giant steps method. Given a cyclic group G with generator g, and given an element t in G, the discrete logarithm problem is that of...

This article gives a gentle introduction to factoring large integers via the quadratic sieve algorithm. The conjectured complexity is worked out in some detail. When faced with a large number n to factor, what do you do first? You might say, "Look at the last digit," with the idea of cheaply pulling out possible factors of 2 and 5. Sure, and more g...

The size of the coefficients of cyclotomic polynomials is a problem that has been well-studied. This paper investigates the following generalization: suppose f (x) ∈ Z[x] is a divisor of x n − 1, so that f (x) is the product of the cyclotomic polynomials corresponding to some of the divisors of n. We ask about the largest coefficient in absolute va...

The size of the coefficients of cyclotomic polynomials is a problem that has been well-studied. This paper investigates the following generalization: suppose $f(x)\in\mathbb{Z}[x]$ is a divisor of $x^n-1$, so that $f(x)$ is the product of the cyclotomic polynomials corresponding to some of the divisors of $n$. We ask about the largest coefficient i...

The quadratic sieve algorithm is currently the method of choice to factor very large composite numbers with no small factors. In the hands of the Sandia National Laboratories team of James Davis and Diane Holdridge, it has held the record for the largest hard number factore since mid-1983. As of this writing, the largest number it has crackd is the...

In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the “number field sieve”, which was proposed by John Pollard. The present paper is devoted to the description and analysis of a more general version of the number field sieve. It should be possible to use this algorithm to factor arbitrary integers into prime fac...

If $a$ is not a multiple of $n$ and $a^{n - 1} \not\equiv 1 \operatorname{mod} n$, then $n$ must be composite and $a$ is called a "witness" for $n$. Let $F(n)$ denote the number of "false witnesses" for $n$, that is, the number of $a \operatorname{mod} n$ with $a^{n - 1} \equiv 1 \operatorname{mod} n$. Considered here is the normal and average size...

Without Abstract

Without Abstract

For a positive integer n and its Euler function (n) we write (n)=n = a=b, where a = a(n) and b = b(n) are coprime. For a xed integer a, we consider the number of integers b for which the above relation holds for some n, and we also x b and count corre- sponding a's. We discuss the greatest common divisor of n and (n), applying it to the relation (n...

We show that an algorithm of V. Miller to compute the group structure of an elliptic curve over a prime finite field runs in probabilistic polynomial time for almost all curves over the field. Important to our proof are estimates for some divisor sums.(Received April 13 2005)

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdos...

Take the product of the numbers (n/(n+1))∊n for 1≤ n < N, where each ∊n is ± 1. Express the product as a/b in lowest terms. Evidently the minimal possible value for a over all choices for ∊n is 1; just take each ∊n = 1, or each ∊n = 0. Denote the maximal possible value of a by A(N). It is known from work of Nicolas and Langevin that (log 4+o(1))N≤...

We give an asymptotic formula for the distribution of those integers n in a residue class, such that n has a fixed sum of base-g digits, with some uniformity over the choice of the modulus and g. We then use this formula to solve the problem of I. Niven of giving an asymptotic formula for the distribution of those integers n divisible by the sum of...

Iteration of the modular l-th power function f(x) = x^l (mod n) provides a common pseudorandom number generator (known as the Blum-Blum-Shub generator when l=2). The period of this pseudorandom number generator is closely related to \lambda(\lambda(n)), where \lambda(n) denotes Carmichael's function, namely the maximal multiplicative order of any i...

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes, the set of products of two primes (of similar size), and the set of all integers. For most $n$ in these sets,...

We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain "smoothing" effect on its integer arguments, our results show that, in fact, most values produced by the Euler function are not smooth. We apply our results to study the distribution of "strong primes", which are...

Let h+('n) denote the class number of the maximal totally real subeld Q(cos(2 ='n)) of the eld of 'n-th roots of unity. The goal of this paper is to show that (speculative extensions of) the Cohen-Lenstra heuristics on class groups provide support for the following conjecture: for all but nitely many pairs ('; n), where ' is a prime and n is a posi...

In this paper we show how to achieve timed fair exchange of digital signatures of standard type. Timed fair exchange (in particular, contract signing) has been considered before, but only for Rabin and RSA signatures of a special kind. Our construction follows the gradual release paradigm, and works on a new “time” structure that we call a mirrored...

In this paper we show how to achieve timed fair exchange of digital signatures of standard type. Timed fair exchange (in particular, contract signing) has been considered before, but only for Rabin and RSA signatures of a special kind. Our construction follows the gradual release paradigm, and works on a new "time" structure that. we call a mirrore...

Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1, then the number {number_sign}(|y|, N) of 1-bits in the expansion of |y| through...