Re: [math-fun] strange factoring in primes
Chris, That's really cool. (But I just wanted to clarify that it was I who asked the question.) Regards, Dan ----------------------------------------------------------------------------- << hihi, all - rich asked about consecutive primes p, q, r with p*q*r = n^3 + 1 for some integer n - i thought i'd take a crack at it without requiring consecutive, and found prety easily 211 * 227 * 241 = 226 ^ 3 + 1 = 11543177, which leadsd me to think that rich's expectation (that only 7*11*13 = 1001 works) is correct BTW, it is easy to show that the middle prime is n+1
In addition to the two examples from previous messages (7 * 11 * 13 = 10^3 + 1 and 211 * 227 * 241 = 226^3 + 1), there are lots more products of three (not necessarily consecutive) primes that make one more than a cube. See http://www.research.att.com/~njas/sequences/A115403 . And, someone wrote, if I understood correctly, that p*q*r = n^3 + 1 implies that q = n + 1, but that's not necessarily true -- IF n +1 is prime, then it's true, but if n+1 is composite (a product of two primes) and n^2 - n + 1 is prime, then there are more possible examples. (Of course in the subcase where p,q,r are consecutive, then the n+1 would have to be prime.) Up to n = 100,00 anyway, there are no more sets of three consecutive primes. There are 946 values of n under 10,000 that yield triples of not-necessarily-consecutive primes (a la A115403). --Joshua Zucker
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Daniel Asimov -
Joshua Zucker