Is there a prime $a^2+b^2$ for all $a \in \mathbb N$?

Question

I am wondering whether it is a known result whether for every natural number $a \geq 1$, there is at least one natural number $b$ (of any size) such that $a^2+b^2$ is prime.

This seems empirically certain, since

1. there are infinitely many primes of the form $a^2+b^2$, and
2. there seems to be at least one prime of form $a^2+b^2$ for every $a$ even with the added constraint that $a \geq b$: see this post: the answer there directed us to a sequnece in OEIS, where it is claimed that with the added constraint, this is an open problem.

Without the added constraint, I am curious whether it's known.

A related conjecture is the Bunyakovsky conjecture, but I am not asking whether there are infinitely many $a^2+b^2$ primes for a fixed $a$, which is very well known to be an open problem.

I am asking whether for every $a\in\mathbb N$, there is at least one $b$ such that $a^2+b^2\in\mathbb P$. This does not preclude the possibility of all integer-valued polynomials having at most finitely many primes, for example.

Answer 1

I printed out a line only when the smallest possible $b$ was strictly larger that previous $b$ values

 a: 8 b: 3  prime  73   b:   3   root a:   2
 a: 12 b: 7  prime  193   b:   7   root a:   3
 a: 30 b: 11  prime  1021   b:   11   root a:   5
 a: 48 b: 13  prime  2473   b:   13   root a:   6
 a: 114 b: 35  prime  14221   b:   35   root a:   10
 a: 408 b: 47  prime  168673   b:   47   root a:   20
 a: 1284 b: 55  prime  1651681   b:   55   root a:   35
 a: 2046 b: 71  prime  4191157   b:   71   root a:   45
 a: 2658 b: 77  prime  7070893   b:   77   root a:   51
 a: 3846 b: 79  prime  14797957   b:   79   root a:   62
 a: 3954 b: 89  prime  15642037   b:   89   root a:   62
 a: 8652 b: 97  prime  74866513   b:   97   root a:   93
 a: 10962 b: 103  prime  120176053   b:   103   root a:   104
 a: 13494 b: 109  prime  182099917   b:   109   root a:   116
 a: 14436 b: 115  prime  208411321   b:   115   root a:   120
 a: 29724 b: 119  prime  883530337   b:   119   root a:   172
 a: 33894 b: 131  prime  1148820397   b:   131   root a:   184
 a: 34062 b: 137  prime  1160238613   b:   137   root a:   184
 a: 35286 b: 139  prime  1245121117   b:   139   root a:   187
 a: 40446 b: 151  prime  1635901717   b:   151   root a:   201
 a: 51714 b: 161  prime  2674363717   b:   161   root a:   227
 a: 75966 b: 175  prime  5770863781   b:   175   root a:   275
 a: 96894 b: 179  prime  9388479277   b:   179   root a:   311
 a: 136260 b: 181  prime  18566820361   b:   181   root a:   369
 a: 138768 b: 205  prime  19256599849   b:   205   root a:   372
 a: 290346 b: 235  prime  84300854941   b:   235   root a:   538
 a: 392184 b: 251  prime  153808352857   b:   251   root a:   626
 a: 562182 b: 293  prime  316048686973   b:   293   root a:   749
 a: 601398 b: 353  prime  361679679013   b:   353   root a:   775