Carl Pomerance

Residue Classes Free of Values of Euler's Function

Dedicated to Andrzej Schinzel on his sixtieth birthday

Irreducible radical extensions and Euler-function chains

Proceedings of the 'Integers Conference 2005' in Celebration of the 70th Birthday of Ronald Graham, Carrollton, Georgia, October 27-30, 2005, 2007

We discuss the smallest algebraic number field which contains the nth roots of unity and which ma... more We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is ϕ(m), where m is the smallest multiple of n divisible by each prime factor of ϕ(m). The prime factors of m/n are precisely the primes not dividing n but which do divide some number in the "Euler chain" ϕ(n), ϕ(ϕ(n)), . . . . For each fixed k, we show that D(n) > n k on a set of asymptotic density 1.

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On pseudosquares and pseudopowers

Introduced by Kraitchik and Lehmer, an x-pseudosquare is a positive integer n ≡ 1 (mod 8) that is... more Introduced by Kraitchik and Lehmer, an x-pseudosquare is a positive integer n ≡ 1 (mod 8) that is a quadratic residue for each odd prime p ≤ 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 power of g yet is so modulo p for all primes p ≤ x. It is conjectured by Bach, Lukes, Shallit, and Williams that the least such number is at most exp(a g x/ log x) for a suitable constant a g . A bound of exp(a g x log log x/ log x) is proved conditionally on the Riemann Hypothesis for Dedekind zeta functions, thus improving on a recent conditional exponential bound of Konyagin and the present authors. We also give a GRH-conditional equidistribution result for pseudopowers that is analogous to our unconditional result for pseudosquares.

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Two contradictory conjectures concerning Carmichael numbers

Mathematics of Computation, 2002

Erd} os conjectured that there are x1 o(1) Carmichael numbers up to x, whereas Shanks was skeptic... more Erd} os conjectured that there are x1 o(1) Carmichael numbers up to x, whereas Shanks was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance showed that there are more than x2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks

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On the Range of the Iterated Euler Function

For a positive integer k letk be the k-fold composition of the Euler function `. In this paper, w... more

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On the Problem of Uniqueness for the Maximum Stirling Number(S) of the Second Kind

Say that an integer n is exceptional if the maximum Stirling number of the second kind S(n;k) occ... more

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The distribution of smooth numbers in arithmetic progressions

Proceedings of The American Mathematical Society - PROC AMER MATH SOC, 1992

We estimate the number of integers n up to x in the arithmetic progression a(mod^) with n free of... more

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Rigorous discrete logarithm computations in nite elds via smooth polynomials

On the Distribution in Residue Classes of Integers with a Fixed Sum of Digits

The Ramanujan Journal, 2005

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Lower Bounds on the Period of Some Pseudorandom Number Generators

1 2 When we refer to the Generalized Riemann Hypothesis in this note we shall mean the Riemann Hy... more

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A Pipeline Architecture for Factoring Large Integers with the Quadratic Sieve Algorithm

SIAM Journal on Computing, 1988

We describe the quadratic sieve factoring algorithm and a pipeline architecture on which it could... more We describe the quadratic sieve factoring algorithm and a pipeline architecture on which it could be efficiently implemented. Such a device would be of moderate cost to build and would be able to factor 100-digit numbers in less than a month. This represents an order of magnitude speed-up over current implementations on supercomputers. Using a distributed network of many such devices, it is predicted much larger numbers could be practically factored.

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On a Conjecture of R.L. Graham

Rocky Mountain Journal of Mathematics, 1994

$Research paper thumbnail of On the normal number of prime factors of $\phi(n)$$

On the normal number of prime factors of $\phi(n)$

Rocky Mountain Journal of Mathematics, 1985

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On a problem of Oppenheim concerning “factorisatio numerorum”

by Carl Pomerance and E. Canfield

Journal of Number Theory, 1983

ABSTRACT

Error estimates for the Davenport-Heilbronn theorems

Duke Mathematical Journal, 2010

We improve the known remainder terms for the theorems of Davenport and Heilbronn on the density o... more

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On some problems of Mąkowski–Schinzel and Erdős concerning the arithmetical functions φ and σ

Colloquium Mathematicum, 2002

Let σ(n) denote the sum of positive divisors of the integer n, and let φ denote Euler's function,... more Let σ(n) denote the sum of positive divisors of the integer n, and let φ denote Euler's function, that is, φ(n) is the number of integers in the interval [1, n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(φ(n))/n ≥ 1/2 for all n. We show that σ(φ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(φ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that φ(n − φ(n)) < φ(n) on a set of asymptotic density 1.

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On the Distribution of Pseudopowers

by Sergei Konyagin and Carl Pomerance

Canadian Journal of Mathematics, 2010

An x-pseudopower to base g is a positive integer which is not a power of g yet is so modulo p for... more 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. 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.

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On locally repeated values of certain arithmetic functions. II

Acta Mathematica Hungarica, 1987

The iterated Carmichael λ-function and the number of cycles of the power generator

Acta Arithmetica, 2005

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On the Number of Distinct Values of Euler''S''Function

by H. Kornmaier and Carl Pomerance

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