A379744 Number of primitive Pythagorean quintuples (a, b, c, d, e) with 0 < a <= b <= c <= d <= e <= 10^n.

10, 5568, 5302303, 5279762116, 5277410421368, 5277177914347752, 5277147974562930196, 5277145259376056385184, 5277145005746992952994327

Offset: 1

Author

Asif Ahmed, Dec 31 2024

Keywords

• nonn
• more

Comments

A Pythagorean quintuple (x,y,z,w,u) is a solution to x^2+y^2+z^2+w^2=u^2.

Examples

			a(1) = 10 because there are ten primitive solutions (a, b, c, d, e) as follows: (1, 1, 1, 1, 2), (1, 1, 3, 5, 6), (1, 1, 7, 7, 10), (1, 2, 2, 4, 5), (1, 3, 3, 9, 10), (1, 4, 4, 4, 7), (1, 5, 5, 7, 10), (2, 2, 3, 8, 9), (2, 2, 4, 5, 7), and (2, 4, 5, 6, 9) with e <= 10.
		

Links

• Math Stack Exchange, Pythagorean quintuple x^2+y^2+z^2+w^2=u^2 has infinite solutions. Is it difficult to find the general formula of its integer solution?.

Crossrefs

Cf. A101931, A157085, A225207.

Formula

Limit_{n -> oo} a(n)/ 10^(3*n) = 5/(96*Pi^2) ~ 0.005277144981371758929368722042173314526269...

a(n) ~ 5*10^(3*n)/(96*Pi^2) + (3/A - 1/G)*10^(2*n)/64 + (1/(2*sqrt(3)) - 1/(4*sqrt(2)))*10^n/Pi, where A is the Dirichlet L-function value evaluated at s = 2 for the Dirichlet character with modulus 8 and index 4, and G is the Catalan's constant. (A ~ 1.064734171043503370392827451461668889483, G ~ 0.9159655941772190150546035149323841107741)