OFFSET
1,1
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.
FORMULA
a(n) ≍ (n log n)^(4/3).
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Charles R Greathouse IV, Jan 11 2022
STATUS
approved