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A301515
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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.
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0
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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a(3) = 16: 16^2 - 16 = 240 = 3^5 - 3.
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A102461 (the complete list of solutions x to (x^2 - x)/2 = (y^3 - y)/6).
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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