A220554 Number of ways to write 2n = p+q (q>0) with p, 2p+1 and (p-1)^2+q^2 all prime

0, 2, 3, 2, 2, 2, 2, 3, 3, 3, 1, 1, 2, 3, 3, 1, 2, 3, 4, 3, 4, 2, 2, 2, 3, 1, 3, 3, 5, 3, 1, 2, 2, 2, 5, 2, 1, 2, 2, 5, 1, 2, 4, 3, 4, 4, 3, 5, 4, 4, 1, 2, 2, 2, 4, 4, 4, 4, 6, 6, 4, 2, 6, 4, 4, 4, 2, 2, 5, 6, 3, 2, 3, 5, 5, 4, 3, 2, 4, 4, 2, 4, 4, 4, 4, 3, 4, 3, 5, 6, 3, 4, 5, 5, 3, 1, 2, 5, 3, 4

Offset

1,2

Comments

Conjecture: a(n)>0 for all n>1.

This has been verified for n up to 2*10^8. It implies that there are infinitely many Sophie Germain primes.

Note that Ming-Zhi Zhang asked (before 1990) whether any odd integer greater than 1 can be written as x+y (x,y>0) with x^2+y^2 prime, see A036468.

Zhi-Wei Sun also made the following related conjectures:

(1) Any integer n>2 can be written as x+y (x,y>=0) with 3x-1, 3x+1 and x^2+y^2-3(n-1 mod 2) all prime.

(2) Each integer n>3 not among 20, 40, 270 can be written as x+y (x,y>0) with 3x-2, 3x+2 and x^2+y^2-3(n-1 mod 2) all prime.

(3) Any integer n>4 can be written as x+y (x,y>0) with 2x-3, 2x+3 and x^2+y^2-3(n-1 mod 2) all prime. Also, every n=10,11,... can be written as x+y (x,y>=0) with x-3, x+3 and x^2+y^2-3(n-1 mod 2) all prime.

(4) Any integer n>97 can be written as p+q (q>0) with p, 2p+1, n^2+pq all prime. Also, each integer n>10 can be written as p+q (q>0) with p, p+6, n^2+pq all prime.

(5) Every integer n>3 different from 8 and 18 can be written as x+y (x>0, y>0) with 3x-2, 3x+2 and n^2-xy all prime.

All conjectures verified for n up to 10^9. - Mauro Fiorentini, Sep 21 2023

References

R. K. Guy, Unsolved Problems in Number Theory, 2nd Edition, Springer, New York, 2004, p. 161.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Example

a(16)=1 since 32=11+21 with 11, 2*11+1=23 and (11-1)^2+21^2=541 all prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[p]==True&&PrimeQ[2p+1]==True&&PrimeQ[(p-1)^2+(2n-p)^2]==True, 1, 0], {p, 1, 2n-1}]
Do[Print[n, " ", a[n]], {n, 1, 1000}]

Crossrefs

Cf. A005384, A005385, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

Sequence in context: A338238 A355077 A111497 * A208243 A209320 A097051

Adjacent sequences: A220551 A220552 A220553 * A220555 A220556 A220557

Keyword

nonn

Author

Zhi-Wei Sun, Dec 15 2012

Status

approved