a composite integer n such that for each prime divisor p of n, (p+1)|(n+1).
Similarly, Which integers N have these properties? - N has at least two distinct prime factors. - If a prime P divides N, then (P-1)|(N-1) and (P+1)|(N+1). (The first condition just dispels all the boring prime-powers.) Here are the first few N's: 74431 = 7^4 * 31 71528191 = 7^4 * 31^3 178708831 = 7^8 * 31 125780831 = 11^6 * 71 4150390625 = 5^12 * 17 You might enjoy proving: - 2 doesn't divide any such N. - 3 doesn't divide any such N. - The set of such N's is infinite. - If a prime P divides an N, then P==N mod 12. And I wonder: - Is there such an N with more than two distinct prime factors? - Is there such an N congruent to 1 mod 12? (If not, then all N's have an odd number of (multiply-counted) prime factors.) -- Don Reble djr@nk.ca