A094597 Number of solutions to the Lebesgue-Nagell equation x^2 + n = y^k with k > 2 and unique x.

1, 0, 2, 0, 0, 5, 1, 0, 0, 2, 1, 1, 0, 2, 2, 1, 2, 2, 1, 0, 0, 3, 0, 1, 2, 1, 6, 0, 0, 2, 3, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 2, 0, 5, 2, 2, 0, 0, 0, 2, 1, 2, 2, 0, 0, 0, 4, 1, 0, 3, 2, 1, 0, 1, 0, 0, 0, 3, 2, 0, 2, 0, 2, 1, 0, 2, 1, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 2, 1, 1, 0, 1, 4

Offset

2,3

Comments

Solutions such as 181^2+7 = 32^2 = 8^5 = 2^15 are counted only once. A094596 counts this as three solutions. Bugeaud, Mignotte and Siksek find all solutions for n <= 100.

Links

Table of n, a(n) for n=2..100.

Yann Bugeaud, Maurice Mignotte and Samir Siksek, Classical and modular approaches to exponential Diophantine equations II. The Lebesgue-Nagell equation

Example

a(4) = 2 because there are two solutions: 2^2+4=2^3 and 11^2+4=5^3.

Mathematica

Table[cnt=0; xLst={}; Do[x=Sqrt[y^k-n]; If[IntegerQ[x] && !MemberQ[xLst, x], cnt++; AppendTo[xLst, x]], {k, 3, 20}, {y, 600}]; cnt, {n, 2, 100}]

Crossrefs

Cf. A094596, A094598, A094599.

Sequence in context: A329959 A259701 A147843 * A202992 A158830 A275744

Adjacent sequences: A094594 A094595 A094596 * A094598 A094599 A094600

Keyword

hard,nonn

Author

T. D. Noe, May 13 2004

Status

approved