## S000511

Primitive Pythagorean triples (in reverse order) in which the hypotenuse and a leg are palindromes in base 2.

5, 4, 3, 17, 15, 8, 65, 63, 16, 65, 56, 33, 257, 255, 32, 365, 364, 27, 1025, 1023, 64, 4097, 4095, 128, 16385, 16383, 256, 65537, 65535, 512, 84005, 68876, 48093, 97661, 95060, 22389, 262145, 262143, 1024, 323705, 321104, 40953, 1048577, 1048575, 2048

1

See S000510.

T. D. Noe, Plot of 26 triples

T. D. Noe, Table of 26 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 = 2^10 - 1; 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, 2]; {FromDigits[Join[Most[s], Reverse[s]], 2], FromDigits[Join[s, Reverse[s]], 2]}, {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}]; Sort[t, #1[[3]] < #2[[3]] &]

Cf. S000510, S000512.

nonn,base,hard

T. D. Noe, Feb 26 2015