Further tabulation of the Erdös-Selfridge function

Abstract

Abstract. For a positive integer k, the Erdös-Selfridge function is the least integer g(k) > k + 1 such that all prime factors of g (k) k exceed k. This paper describes a rapid method of tabulating g(k) using VLSI based sieving hardware. We investigate the number of admissible residues for each modulus in the underlying sieving problem and relate this number to the size of g(k). A table of values of g(k) for 135 ≤ k ≤ 200 is provided.

An Algorithm and Estimates for the Erdős-Selfridge Function (work in progress)

“…(1) We present a new algorithm to compute the value of g(k), and use it to both verify previous work [1,16,12] and compute new values of g(k), with our current limit being g(323) = 1 69829 77104 46041 21145 63251 22499.…”

“…This theorem allows one to set up a sieve problem to search for g(k) as the smallest residue, larger than k + 1, modulo M k , that satisfies Kummer's criteria. Lukes, Scheidler, and Williams [12] then improved their sieve, used special-purpose hardware, and computed g(k) for all k ≤ 200.…”

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Let p(n) denote the smallest prime divisor of the integer n. Define the function g(k) to be the smallest integer > k + 1 such that p( g(k) k ) > k. So we have g(2) = 6 and g(3) = g(4) = 7. In this paper we present the following new results on the Erdős-Selfridge function g(k):(1) We present a new algorithm to compute the value of g(k), and use it to both verify previous work [1,16,12] and compute new values of g(k), with our current limit being g(323) = 1 69829 77104 46041 21145 63251 22499.(2) We define a new function ĝ(k), and under the assumption of our uniform distribution heuristic we show that

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“…Note that our algorithm deals with two sub-problems that have both been proven to be NP-complete: the knapsack problem [4] and finding the smallest solution to a system of modular congruences [13]. For previous work on the Erdős-Selfridge function, see [1,2,16,12,11,5].…”

An algorithm and estimates for the Erdős–Selfridge function

“…In fact, we defineĝ(k) := M k /R k . The authors of [11] studied this approximating function; it plays a central role in the analysis of our algorithm, but not in its correctness.…”