Each of A,B and C is a positive integer such that each of the expressions A*B+A+B, B*C+B+C, C*A+C+A, A*B+C and A*C+B is a perfect square.
(I) Find the smallest value of A+B+C.
(II) Does there exist an infinity of triplets (A,B,C) satisfying the given conditions?
Give reasons for your answer.
(In reply to
re: Two small solutions and a conjecture by Brian Smith)
F(n + 1) F(n 1) F(n)^2 = (1)^n (Cassini)
From which: F(2n + 1) F(2n  1) = F(2n)^2 +1
We are going to add an irrelevance, because that's what the problem does: let A = F(2k1)1, and B = F(2k+1)1
We have something like: xy=(z^2+1), so (x1)(y1) = (z^2+1)xy+1, so when we add back A and B in the original equation: (z^2+1)xy+1+(x1)+(y1) we obtain z^2, after cancelling. If it's true of A and B, it has to be true of B and C, if B = F(2k+1)1 and C = F(2k+3)1
Similar manipulation of the other equations produce the squares as required in like manner.
For the others: above we had (z^2+1)xy+1. Now between x and y we have (yx), so (C+1) = y+(y+(yx) = 3yx, giving 2x+2y+z^2+1. But z=(yx), so 2z+z^2+1 = (z+1)^2
Similar manipulation of the other equations produce the squares as required in like manner.
Edited on December 28, 2015, 9:28 am

Posted by broll
on 20151228 04:10:19 