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

Author

Tomohiro Yamada, Dec 15 2018

Keywords

• nonn
• fini
• full

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.

Examples

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

Links

• 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.

Crossrefs

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

Programs

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 *)