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.

3, 6, 16, 91, 280

Offset

1,1

Comments

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.

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).

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).

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.

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.

Links

Table of n, a(n) for n=1..5.

Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll and Szabolcs Tengely, Integral points on hyperelliptic curves, Algebra Number Theory 2 (2008), 859-885.

Daniel C. Fielder and Cecil O. Alford, Observations from computer experiments on an integer equation, Applications of Fibonacci numbers, edited by G. E. Bergum, A. N. Philippou and A. F. Horadam, vol. 7, pp. 93-103.

M. Mignotte and A. Pethő, On the diophantine equation x^p - x = y^q - y, Publ. Mat. 43 (1999), 207-216.

L. J. Mordell, On the integer solutions of y(y+1)=x(x+1)(x+2), Pacific J. Math. 13 (1963), 1347-1351.

Example

a(3) = 16: 16^2 - 16 = 240 = 3^5 - 3.

Mathematica

r[x_, q_] := {x, y, q} /. {ToRules @ Reduce[y >= 2 && x^2 - x == y^q - y, y, Integers]};
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 = Table[r[x], {x, 2, 300}];
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 *)

Crossrefs

Cf. A102461 (the complete list of solutions x to (x^2 - x)/2 = (y^3 - y)/6).

Sequence in context: A028405 A328276 A191223 * A014645 A195996 A036050

Adjacent sequences: A301512 A301513 A301514 * A301516 A301517 A301518

Keyword

nonn,fini,full

Author

Tomohiro Yamada, Dec 15 2018

Status

approved