A349900 Primes of the form x^2 + (y^2+1)^2.

5, 13, 17, 29, 37, 41, 53, 61, 89, 101, 109, 149, 173, 181, 197, 229, 257, 269, 281, 293, 349, 353, 389, 401, 433, 461, 509, 541, 577, 601, 613, 677, 701, 733, 757, 773, 797, 809, 829, 941, 1049, 1061, 1093, 1117, 1181, 1229, 1297, 1301

Offset: 1

Author

Charles R Greathouse IV, Jan 11 2022

Keywords

• nonn

Comments

Merikoski proved that there are infinitely many primes of this form, and that the order of growth of the sequence up to x is x^(3/4)/log x. (His method did not provide enough Type II information to prove that there is a C such that there are ~ C*x^(3/4)/log x.)

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Jori Merikoski, The polynomials X^2+(Y^2+1)^2 and X^2+(Y^3+Z^3)^2 also capture their primes, arXiv:2112.03617 [math.NT], 2021.

Crossrefs

Subsequence of A002144.

Cf. A028916, A350687.

Programs

PARI

list(lim)=my(v=List()); lim\=1; for(y=0,sqrtint(sqrtint(lim-1)-1), my(t=(y^2+1)^2); forstep(x=2-y%2,sqrtint(lim-t),2, my(p=x^2+t); if(isprime(p), listput(v,p)))); Set(v)

PARI

is(n)=if(n<5 || !isprime(n), return(0)); for(y=0,sqrtint(sqrtint(n-1)-1), if(isprime(n-(y^2+1)^2), return(1))); 0

Formula

a(n) ≍ (n log n)^(4/3).