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%I #20 Sep 21 2023 15:36:06
%S 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,
%T 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,
%U 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
%N Number of ways to write 2n = p+q (q>0) with p, 2p+1 and (p-1)^2+q^2 all prime
%C Conjecture: a(n)>0 for all n>1.
%C This has been verified for n up to 2*10^8. It implies that there are infinitely many Sophie Germain primes.
%C 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.
%C Zhi-Wei Sun also made the following related conjectures:
%C (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.
%C (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.
%C (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.
%C (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.
%C (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.
%C All conjectures verified for n up to 10^9. - _Mauro Fiorentini_, Sep 21 2023
%D R. K. Guy, Unsolved Problems in Number Theory, 2nd Edition, Springer, New York, 2004, p. 161.
%H Zhi-Wei Sun, <a href="/A220554/b220554.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2017.
%e a(16)=1 since 32=11+21 with 11, 2*11+1=23 and (11-1)^2+21^2=541 all prime.
%t 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}]
%t Do[Print[n," ",a[n]],{n,1,1000}]
%Y Cf. A005384, A005385, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Dec 15 2012
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