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
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
KEYWORD
nonn,fini,full
AUTHOR
Tomohiro Yamada, Dec 15 2018
STATUS
approved