84.41 An interesting conjecture

Diophantine triples

“…If we take k = 1 then S" = F n (the Fibonacci sequence) and if we now use (5), and note that F n + 2F n+l + F n+2 = (F n + F n+ i) + (F n+ i + F n+2 ) = F n+2 + F" +3 = F" +4 , we recapture the fact that (F 2m , F 2m + 2 , F 2m + 4) is a Diophantine triple. When k = 2 we have S n = P n (the Pell sequence), and here (5) implies that (p2m,P2m+2, IPim+i) is a Diophantine triple. In this case we also have k = 2£, where 6 = 1, and so from (6), we see that (P 2m , 2P 2m , 5P 2m + 2P 2m -i) is a Diophantine triple.…”

“…The we have the following result (in which the special case (a, b, c) = (0, 2, 4) was considered in [5]). (a, b, c) is a Diophantine triple with ab+1 = q 2 , c = a + b + 2q,c> a + b and b even.…”

“…It was conjectured in [5] (and proved there by the then Editor) that if x = (0, 2, 4), then x (A T )" is a Diophantine triple for every positive n, where A T denotes the transpose of A. In this note we shall use a little linear algebra to explain why this (and much more) is true, and to show how to generate other sequences of Diophantine triples by using recurrence relations.…”

“…This is since in this case a 5 ' -i' = {a s -i)(fl s( ' _1) + i« l( '-2) + i V ' -3 ) +... + i ( ' " V + I*""). Now, if a 1 + 1 is prime it is clear that a must be even.…”