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Numbers whose square can be expressed as the signed sum of a fifth power and a cube: z^2 = x^5 + y^3 with gcd(x,y,z)=1.
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%I #13 Jul 08 2023 18:30:55

%S 3,10,411,654,7792,36599,39151,647992,1506463,1525899,2730128,3353687,

%T 4387861,4942947,5574720,12092581,128301258,168454745,184589480,

%U 888155653,20364997771,53242416249,65464918703,73699708330,74330984303

%N Numbers whose square can be expressed as the signed sum of a fifth power and a cube: z^2 = x^5 + y^3 with gcd(x,y,z)=1.

%H Dario Alpern, <a href="https://www.alpertron.com.ar/SPOW532.HTM">Sum of powers a^5 + b^3 = c^2.</a>

%H Johnny Edwards, <a href="https://doi.org/10.1515/crll.2004.043">A Complete Solution to X^2+Y^3+Z^5=0.</a> Journal für die reine und angewandte Mathematik (Crelle's Journal) 571, 213-236 (2004).

%e a(1)=3 because 1^5 + 2^3 = 3^2;

%e a(2)=10 because (-3)^5 + 7^3 = 10^2;

%e a(3)=411 because 10^5 + 41^3 = 411^2;

%e a(4)=654 because 19^5 + (-127)^3 = 654^2.

%Y Cf. A070065 positive integer solutions of x^2 + y^5 = z^3.

%K nonn

%O 1,1

%A _Hugo Pfoertner_, Jan 25 2005