A290935 Number of ways to write 2*n+1 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x^2 + 3*y^2 + 5*z^2 + 7*w^2 and p - 2 are twin prime.

2, 1, 4, 2, 1, 1, 4, 2, 2, 6, 2, 1, 6, 1, 2, 8, 5, 3, 7, 1, 4, 10, 3, 2, 9, 4, 8, 7, 5, 5, 11, 4, 7, 8, 4, 3, 10, 5, 6, 10, 7, 4, 16, 4, 9, 10, 2, 3, 11, 7, 5, 8, 3, 7, 13, 4, 4, 16, 2, 6, 15, 1, 4, 10, 6, 6, 13, 7, 2, 13, 8, 9, 15, 4, 12, 8, 7, 5, 7, 2, 9

Offset

0,1

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 1, 4, 5, 11, 13, 19, 61.

This refinement of Lagrange's four-square theorem implies the twin prime conjecture.

Below we list some similar conjectures:

(i) Any positive odd integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = 3*x^2 + 5*y^2 + 11*z^2 + 13*w^2 and p + 2 are twin prime.

(ii) Each positive odd integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x + y + 3*z + 5*w and p + 2 (or p - 2) are twin prime.

(iii) Any positive odd integer can be can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^3 + y^3 + z^3 + 3*w^3 is prime.

(iv) For each m = 1,2,3, any positive integer not divisible by 4 can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^m + 2*y^m + 3*z^m + 4*w^m is prime.

(v) Let n be any positive integer not divisible by 4. Then we can write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 2*x^4 + 3*y^4 + 4*z^4 + 5*w^4 is prime. Also, we can write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 3*x^5 + 4*y^5 + 5*z^6 + 6*w^5 is prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..5000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Example

a(0) = 2 since 2*0+1 = 0^2 + 0^2 + 1^2 + 0^2 with 0^2 + 3*0^2 + 5*1^2 + 7*0^2 = 5 and 5 - 2 = 3 twin prime, and 2*0+1 = 0^2 + 0^2 + 0^2 + 1^2 with 0^2 + 3*0^2 + 5*0^2 + 7*1^2 = 7 and 7 - 2 = 5 prime.
a(1) = 1 since 2*1+1 = 1^2 + 0^2 + 1^2 + 1^2 with 1^2 + 3*0^2 + 5*1^2 + 7*1^2 = 13 and 13 - 2 = 11 twin prime.
a(4) = 1 since 2*4+1 = 2^2 + 0^2 + 2^2 + 1^2 with 2^2 + 3*0^2 + 5*2^2 + 7*1^2 = 31 and 31 - 2 = 29 twin prime.
a(5) = 1 since 2*5+1 = 3^2 + 1^2 + 0^2 + 1^2 with 3^2 + 3*1^2 + 5*0^2 + 7*1^2 = 19 and 19 - 2 twin prime.
a(11) = 1 since 2*11+1 = 3^2 + 2^2 + 3^2 + 1^2 with 3^2 + 3*2^2 + 5*3^2 + 7*1^2 = 73 and 73 - 2 = 71 twin prime.
a(13) = 1 since 2*13+1 = 1^2 + 0^2 + 1^2 + 5^2 with 1^2 + 3*0^2 + 5*1^2 + 7*5^2 = 181 and 181 - 2 = 179 twin prime.
a(19) = 1 since 2*19+1 = 1^2 + 3^2 + 5^2 + 2^2 with 1^2 + 3*3^2 + 5*5^2 + 7*2^2 = 181 and 181 - 2 = 179 twin prime.
a(61) = 1 since 2*61+1 = 7^2 + 3^2 + 7^2 + 4^2 with 7^2 + 3*3^2 + 5*7^2 + 7*4^2 = 433 and 433 - 2 = 431 twin prime.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
TQ[p_]:=TQ[p]=PrimeQ[p]&&PrimeQ[p-2];
Do[r=0; Do[If[SQ[2n+1-x^2-y^2-z^2]&&TQ[x^2+3y^2+5z^2+7(2n+1-x^2-y^2-z^2)], r=r+1], {x, 0, Sqrt[2n+1]}, {y, 0, Sqrt[2n+1-x^2]}, {z, 0, Sqrt[2n+1-x^2-y^2]}]; Print[n, " ", r], {n, 0, 80}]

Crossrefs

Cf. A000040, A000118, A000290, A001359, A006512, A271518, A281976, A291150, A291191.

Sequence in context: A214027 A007739 A330086 * A031424 A013942 A187816

Adjacent sequences: A290932 A290933 A290934 * A290936 A290937 A290938

Keyword

nonn

Author

Zhi-Wei Sun, Aug 23 2017

Status

approved