## S000579

Primitive Pythagorean triples in which the two legs and the hypotenuse are palindromes in base 9, shown in base 10.

3, 4, 5, 60, 91, 109, 359160, 624439, 720361

1

See S000580 for the palindromes.

T. D. Noe, Plot of 3 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) PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; base = 9; nn = base^4 - 1; pal = Table[s = IntegerDigits[n, base]; {FromDigits[Join[Most[s], Reverse[s]], base], FromDigits[Join[s, Reverse[s]], base]}, {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]; s = IntegerDigits[z, base]; If[z <= pal[[-1]] && s == Reverse[s], AppendTo[t, {x, y, z}]]], {x, pal}, {y, pal}]; t = Sort[t, #1[] < #2[] &]

Cf. S000501-S000509S000580.

nonn,base,hard

T. D. Noe, Apr 15 2015