## S000502

Primitive Pythagorean triples (in reverse order) in which the two legs are palindromes.

5, 4, 3, 905, 777, 464, 48985, 48984, 313, 55145, 42824, 34743, 61373, 55755, 25652, 96433, 80808, 52625, 2228137, 2152512, 575575, 5579333, 5578755, 80308, 6377873, 5853585, 2532352, 24382493, 23711732, 5679765, 504110405, 402080204, 304070403

1

The triples are shown in reverse sorted order: z, y, x with x^2 + y^2 = z^2. The numbers 777 and 464 are the palindromic legs of the second triple.

T. D. Noe, Plot of 14 triples

T. D. Noe, Table of 14 triples

John Rafael M. Antalan and Richard P. Tagle, Numeric palindromes in primitive and non-primitive Pythagorean triples, arxiv.org 1502.06760 (Feb 24 2015)

Eric W. Weisstein, MathWorld: Pythagorean Triple

(Mma) nn = 999; PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; pal = Table[s = IntegerDigits[n]; {FromDigits[Join[Most[s], Reverse[s]]], FromDigits[Join[s, Reverse[s]]]}, {n, nn}]; pal = Sort[Flatten[pal]]; t = {}; Do[If[x < y && GCD[x, y] == 1 && PerfectSquareQ[x^2 + y^2], z = Sqrt[x^2 + y^2]; If[z <= pal[[-1]], AppendTo[t, {x, y, z}]]], {x, pal}, {y, pal}]; Reverse /@ Sort[t, #1[[3]] < #2[[3]] &]

Cf. S000501-S000509S000579S000580.

nonn,base,hard

T. D. Noe, Feb 25 2015

© Tony D Noe 2014-2015