The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 0
 3, 6, 16, 91, 280 (list; graph; refs; listen; history; text; internal format)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 9 03:23 EST 2021. Contains 349625 sequences. (Running on oeis4.)