A060457 Number of solutions to y^2 + y = x^3 - x^2 modulo n.

1, 4, 4, 8, 4, 16, 9, 16, 12, 16, 10, 32, 9, 36, 16, 32, 19, 48, 19, 32, 36, 40, 24, 64, 20, 36, 36, 72, 29, 64, 24, 64, 40, 76, 36, 96, 34, 76, 36, 64, 49, 144, 49, 80, 48, 96, 39, 128, 63, 80, 76, 72, 59, 144, 40, 144, 76, 116, 54, 128, 49, 96, 108, 128, 36, 160, 74

Offset: 1

Author

Frank Ellermann, Apr 09 2001

Keywords

• nonn
• mult

Comments

Singh mistakenly called this the L-series, but the L-series for elliptic curve y^2 + y = x^3 - x^2 is A006571. - Michael Somos, Mar 20 2010

Examples

			a(5)=4 from the 4 solutions (0,0), (0,4), (1,0), (1,4) mod 5.
G.f. = x + 4*x^2 + 4*x^3 + 8*x^4 + 4*x^5 + 16*x^6 + 9*x^7 + 16*x^8 + 12*x^9 + ...
		

References

• Simon Singh, Fermat's last theorem, 1997 (at the end of ch. 4).

Links

• Robin Visser, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A061011, A061012.

Programs

PARI

{a(n) = sum(x=0, n-1, sum(y=0, n-1, (y^2 + y - x^3 + x^2) % n == 0))}; /* Michael Somos, Mar 20 2010 */

PARI

{a(n) = local(E, A, p, e); if(n<1, 0, E = ellinit( [0, -1, 1, 0, 0], 1); A = factor(n); prod( k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; (p - ellap(E, p)) * p^(e-1) )))}; /* Michael Somos, Mar 20 2010 */

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2001