Consecutive Non-Prime Integers
Date: 2/17/96 at 12:42:32 From: Anonymous Subject: help in abstract algebra For any integer n>1 there exist n consecutive non prime integers. I have taken n=4 and then use 24,25,26,27 as a specific example. However, for n large I don't see how this can hold. Can you please send me some insight on the general case?
Date: 3/24/96 at 12:12:30 From: Doctor Steven Subject: Re: help in abstract algebra This is a tough problem but one that has a relatively simple solution once you know it (until then it'll keep you stumped for days). Take a number n>1. Then looking at a_k = (n+1)! + k + 1, where k goes from 1 to n, you get: a_1 = (n+1)! + 2, a_2 = (n+1)! + 3, : : a_n = (n+1)! + n + 1. Notice we have n consecutive integers. Notice 2, 3, ...., n+1 all divide (n+1)!. Notice also that 2, 3, (n+1) divide 2, 3, ... (n+1) respectively. So we have n consecutive composite integers. -Doctor Steven, The Math Forum
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