Primitive Pythagorean triples in which the hypotenuse and a leg are palindromes.
3, 4, 5, 20, 99, 101, 225, 272, 353, 252, 275, 373, 33, 544, 545, 468, 595, 757, 555, 572, 797, 1551, 9880, 10001, 200, 9999, 10001, 2020, 10101, 10301, 2220, 12221, 12421, 2420, 14541, 14741, 8668, 38325, 39293, 30393, 42224, 52025, 31007, 42024, 52225
1
The triples are shown sorted: x, y, z with x^2 + y^2 = z^2. The numbers 99 and 101 are the palindromic leg and hypotenuse of the second triple. This sequence is infinite because it contains the infinite subsequence of triples 2*10^k, 10^(2k)-1, 10^(2k)+1 for k > 0.
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}]; Sort[t, #1[[3]] < #2[[3]] &]
Cf. S000501-S000509, S000579, S000580.
nonn,base,hard,nice
T. D. Noe, Feb 25 2015