A127308 Number of ways of writing the n-th prime prime(n) as a sum of 24 squares.

1104, 16192, 1362336, 44981376, 6631997376, 41469483552, 793229226336, 2697825744960, 22063059606912, 282507110257440, 588326886375936, 4119646755044256, 12742799887509216, 21517654506205632, 57242599902057216

Offset: 1

Author

Jonathan Sondow, Jan 10 2007

Keywords

• nonn

Comments

|a(n) - (16/691)*(prime(n)^11 + 1)| <= (66304/691)*sqrt(prime(n)^11) (proved by Deligne).

Examples

			For prime(1) = 2, two of the 24 squares are (+-1)^2 and the other 22 are 0^2, so a(1) = 2*2*binomial(24,2) = 4*276 = 1104.
		

References

• E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 107.
• Barry Mazur, Controlling our errors, Nature 443, 7 (2006) 38-40.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..70
• Barry Mazur, The Structure of Error Terms in Number Theory and an Introduction to the Sato-Tate Conjecture, Current Events Bulletin, Amer. Math. Soc., 2007.
• Barry Mazur, Controlling our errors
• Barry Mazur, Finding meaning in error terms, Bull. Amer. Math. Soc., 45 (No. 2, 2008), 185-228.
• Tony Phillips, Math in the Media

Crossrefs

Cf. A000040, A000156, A000594.

Programs

Mathematica

Table[SquaresR[24, Prime[n]], {n, 1, 70}]
Table[Abs[16/691 (p^11 + 1) + 33152/691 RamanujanTau[p]], {p, Prime@Range@70}] (* Giorgos Kalogeropoulos, Dec 15 2022 *)

Formula

a(n) = A000156(prime(n)).

a(n) ~ (16/691)*(prime(n)^11 + 1) as n -> oo.

a(n) = (16/691)*(prime(n)^11+1) + (33152/691)*tau(prime(n)) for n>1 where tau = A000594. - Giorgos Kalogeropoulos, Dec 15 2022