An Algorithm and Estimates for the Erdős-Selfridge Function (work in progress)
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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): We present a new algorithm to compute the value of g(k), and use it to both verify previous work and compute new values of g(k), with our current limit being g(272)=57 61284 34192 78614 55093 37498. We define a new function ĝ(k), and under the assumption of our Uniform Distribution Heuristic we show that g(k) = ĝ(k) + O( k) with high "probability". We also provide computational evidence to support our claim that ĝ(k) estimates g(k) reasonably well in practice. There are several open conjectures on the behavior of g(k) which we are able to prove for ĝ(k), namely that 1- 2/2+o(1) < ĝ(k)/k/ k < 2+o(1), and that _k→∞ĝ(k+1)/ĝ(k)=∞. Let G(x,k) count the number of integers n< x such that p(nk)>k. Unconditionally, we prove that for large x, G(x,k) is asymptotic to x/ĝ(k). And finally, we show that the running time of our new algorithm is at most g(k) [ -c (k k) /( k)^2 (1+o(1))] for a constant c>0.
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