## S000504

Primitive Pythagorean triples (in reverse order) in which the hypotenuse and a leg are palindromes.

5, 4, 3, 101, 99, 20, 353, 272, 225, 373, 275, 252, 545, 544, 33, 757, 595, 468, 797, 572, 555, 10001, 9880, 1551, 10001, 9999, 200, 10301, 10101, 2020, 12421, 12221, 2220, 14741, 14541, 2420, 39293, 38325, 8668, 52025, 42224, 30393, 52225, 42024, 31007

1

The triples are shown in reverse sorted order: z, y, x with x^2 + y^2 = z^2. The numbers 101 and 99 are the palindromic hypotenuse and leg of the second triple.

T. D. Noe, Plot of 58 triples

T. D. Noe, Table of 58 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[z > x && GCD[x, z] == 1 && PerfectSquareQ[z^2 - x^2], y = Sqrt[z^2 - x^2]; AppendTo[t, Sort[{x, y, z}]]], {z, pal}, {x, pal}]; Reverse /@ Sort[t, #1[[3]] < #2[[3]] &]

Cf. S000501-S000509S000579S000580.

nonn,base,hard

T. D. Noe, Feb 25 2015