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June 7[edit]

An elementary approach for an elementary question.[edit]

For the eqn. ab + cd = t where a,b,c,d are four consecutive numbers. show that "t " will never be a perfect power?

The lhs is equal to 2 modulo 4, which perfect powers are not. --Lambiam 10:42, 7 June 2021 (UTC)[reply]

It depends on what you mean by "consecutive". If it means (a, b, c, d) = (a, a+1, a+2, a+3) then yes, the problem is trivial. But it might mean as sets {a, b, c, d} = {n, n+1, n+2, n+3} for some n. Then there are three essentially different cases, of which 2 are trivial and n(n+2)+(n+1)(n+3)=2n²+6n+3=m^k isn't (as far as I can tell). Though you can reduce mod 8 to prove k is not even. I'm pretty well convinced this is homework; only a homework problem would be so vaguely worded :) The same wording is used in one of the problems above and I didn't realize it was ambiguous when I gave my response to it. --RDBury (talk) 11:36, 7 June 2021 (UTC)[reply]