A227078 The Ramanujan-Nagell squares: A038198(n)^2.

1, 9, 25, 121, 32761

Offset: 0

Author

Raphie Frank, Jun 30 2013

Keywords

• nonn
• fini
• full

Comments

a(n) = (2*x - 1)^2 = (sqrt(2)*sqrt(sqrt(6*y^2 - 5) + 1) - 1)^2 = 2^(z + 3) - 7 for x, y, z are the solutions to two Diophantine equations noted by R. K. Guy: 2*x^2*(x^2 - 1) = 3*(y^2 - 1) & x*(x - 1)/2 = 2^z - 1 (see A180445). x = {1, 2, 3, 6, 91} = A180445(n), y = {1, 3, 7, 29, 6761} = A227077(n), and z = {0, 1, 2, 4, 12} = A215795(n).

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008.
• L. J. Mordell, Diophantine Equations, Academic Press, NY, 1969, p. 205.

Links

• Curtis Bright, Solving Ramanujan's Square Equation Computationally
• Eric Weisstein's World of Mathematics, Ramanujan's Square Equation

Crossrefs

Cf. A060728, A076046, A180445, A227077, A215795.

Formula

a(n) + 7 = 2^A060728(n).

(a(n) - 1)/8 = A076046(n).