Let $s(\cdot )$ denote the sum-of-proper-divisors function, that is, $s(n)=\sum _{d\mid n,~d<n}d$. Erdős, Granville, Pomerance, and Spiro conjectured that for any set $\mathscr{A}$ of asymptotic density zero, the preimage set $s^{-1}(\mathscr{A})$ also has density zero. We prove a weak form of this conjecture: if $\unicode[STIX]{x1D716}(x)$ is any function tending to $0$ as $x\rightarrow \infty$, and $\mathscr{A}$ is a set of integers of cardinality at most $x^{1/2+\unicode[STIX]{x1D716}(x)}$, then the number of integers $n\leqslant x$ with $s(n)\in \mathscr{A}$ is $o(x)$, as $x\rightarrow \infty$. In particular, the EGPS conjecture holds for infinite sets with counting function $O(x^{1/2+\unicode[STIX]{x1D716}(x)})$. We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D716}$, there are integers $n$ with arbitrarily many $s$-preimages lying between $\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D716})n$ and $\unicode[STIX]{x1D6FC}(1+\unicode[STIX]{x1D716})n$. Finally, we make some remarks on solutions $n$ to congruences of the form $\unicode[STIX]{x1D70E}(n)\equiv a~(\text{mod}~n)$, proposing a modification of a conjecture appearing in recent work of the first two authors. We also improve a previous upper bound for the number of solutions $n\leqslant x$, making it uniform in $a$.

We give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.

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 standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

An $x$-pseudopower to base $g$ is a positive integer that 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 behaviour of the multiplicative order of $g$ modulo prime numbers.

Assuming a conjecture intermediate in strength between one of Chowla and one 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 natural number n, let λ(n) denote the order of the largest cyclic subgroup of (ℤ/nℤ)*. 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 (ℤ/nℤ)*. Let R(n) denote the number of elements of (ℤ/nℤ)* with order λ(n). It is natural to compare Na(x) with ∑n≤xR(n)/n. In this paper we show that the average of Na(x) for 1 ≤ a ≤ y is indeed asymptotic to this sum, provided y ≥ exp((2 + ε)(log x log log x)1/2), thus improving a theorem of the first author who had this for y ≥ exp((log x)3/4;). The result is to be compared with a similar theorem of Stephens who considered the case of prime numbers 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.

This series of papers presents and rigorously analyzes a probabilistic algorithm for finding small prime factors of an integer. The algorithm uses the Jacobian varieties of curves of genus 2 in the same way that the elliptic curve method uses elliptic curves. This second paper in the series is concerned with the order of the group of rational points on the Jacobian of a curve of genus 2 defined over a finite field. We prove a result on the distribution of these orders.

2000 Mathematical Subject Classification: 11Y05, 11G10, 11M20, 11N25.

Let Ω be a non-empty open set in ℝn with finite ‘volume’ (n-dimensional Lebesgue measure). Let be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions):

where λ ∈ ℝ and u is a non-zero member of (the closure in the Sobolev space H1(Ω) of the set of smooth functions with compact support contained in Ω). It is well known that the values of λ∈ℝ for which (1·1) has a non-zero solution are positive and form a discrete set. Moreover, for each λ, the associated eigenspace is finite dimensional. Let the spectrum of (1·1) be denoted where 0 < λ1 ≤ λ2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let

For each natural number m, let N(m) denote the number of integers n with ø(n) = m, where ø denotes Euler's function. There are many interesting problems connected with the function N(m), such as the conjecture of Carmichael that N(m) is never 1 (see [9], for example) and the study of the distribution of the m for which N(m) > 0 (see Erdős and Hall [5]). In this note we shall be concerned with the maximal order of N(m).

In this paper we prove

Theorem 1. If N is a natural number and I is an interval of N consecutive integers, then there is a 1–1 correspondencef: {1, 2,…, N} → I such that (i, f(i)) = 1 for 1 ≤ i ≤ N.

The first problem in diophantine approximation is for a given real number ξ, and positive number x to find a fraction s/t with t ≤ x which is close to ξ. This problem can be rephrased in geometric terms. Given a vector v in ℝ2, find a vector a = (a1, a2) with integer coordinates and 1 1 ≤ al ≥ x such that the vectors a and v are nearly parallel. Simultaneous approximation of d − 1 real numbers can be recast in terms of approximation of the angle between two vectors in d-dimensional Euclidean space.