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A301515 Complete list of integers x > 1 such that x^2 - x = y^q - y, where q is an odd prime and y is a prime power.

%I A301515 #16 Dec 17 2018 09:32:23
%S A301515 3,6,16,91,280
%N A301515 Complete list of integers x > 1 such that x^2 - x = y^q - y, where q is an odd prime and y is a prime power.
%C A301515 The corresponding values of (y, q) are (2, 3), (2, 5), (3, 5), (2, 13) and (5, 7). Mignotte and Pethő proved that the list is complete.
%C A301515 If we relax the condition that y should be a prime power, the equation x^2 - x = y^q - y has additionally two solutions (x, y, q) = (15, 6, 3) and (4930, 30, 5) (Fielder and Alford, 1998).
%C A301515 The result of Mordell (1963) implies that x^2 - x = y^3 - y has only three positive integral solutions (x, y) = (1, 1), (3, 2) and (15, 6).
%C A301515 Bugeaud, Mignotte, Siksek, Stoll and Tengely proved that (x, y) = (1, 1), (6, 2), (16, 3), (4930, 30) are the only positive integral solutions to x^2 - x = y^5 - y.
%C A301515 The equation x^p - x = y^q - y, with p, q odd primes and x,y > 1 has a solution 13^3 - 13 = 3^7 - 3 but no other solution is known.
%H A301515 Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll and Szabolcs Tengely, <a href="https://doi.org/10.2140/ant.2008.2.859">Integral points on hyperelliptic curves</a>, Algebra Number Theory 2 (2008), 859-885.
%H A301515 Daniel C. Fielder and Cecil O. Alford, <a href="https://doi.org/10.1007/978-94-011-5020-0_13">Observations from computer experiments on an integer equation</a>, Applications of Fibonacci numbers, edited by G. E. Bergum, A. N. Philippou and A. F. Horadam, vol. 7, pp. 93-103.
%H A301515 M. Mignotte and A. Pethő, <a href="https://doi.org/10.5565/PUBLMAT_43199_08">On the diophantine equation x^p - x = y^q - y</a>, Publ. Mat. 43 (1999), 207-216.
%H A301515 L. J. Mordell, <a href="https://doi.org/10.2140/pjm.1963.13.1347">On the integer solutions of y(y+1)=x(x+1)(x+2)</a>, Pacific J. Math. 13 (1963), 1347-1351.
%e A301515 a(3) = 16: 16^2 - 16 = 240 = 3^5 - 3.
%t A301515 r[x_, q_] := {x, y, q} /. {ToRules @ Reduce[y >= 2 && x^2 - x == y^q - y, y, Integers]};
%t A301515 r[x_] := Select[Table[r[x, q], {q, NextPrime[Log[2, x^2 - x + 2]]}] /. {{a_, b_, c_}} -> {a, b, c}, PrimeNu[#[[2]]]==1 && #[[3]] > 2&];
%t A301515 T = Table[r[x], {x, 2, 300}];
%t A301515 For[k = 1, k <= Length[T], k++, t = T[[k]]; If[t != {}, Print["x = ", t[[1, 1]], ", y = ", t[[1, 2]], ", q = ", t[[1, 3]]]]] (* _Jean-François Alcover_, Dec 17 2018 *)
%Y A301515 Cf. A102461 (the complete list of solutions x to (x^2 - x)/2 = (y^3 - y)/6).
%K A301515 nonn,fini,full
%O A301515 1,1
%A A301515 _Tomohiro Yamada_, Dec 15 2018