6
$\begingroup$

During my experiments with "Mathematica" I arrived to the following observations. My question is that are they interesting, known, solved or not. If they are known could you please give me a reference.

1) Every odd positive integer $n$ can be partitioned in two positive integers $n=a+b$ such that $a^2+b^2$ is prime.

2) For every positive integer $a$ there exists positive integer $b$ such that $a^2+b^2$ is prime.

3) Let $n$ be a positive integer. Then there is a positive integer $a$ such that the number of $b$'s such that $0<b<a$ and $a^2+b^2$ being prime, is $n$.

Thank you.

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ The first question is probably as hard as binary Goldbach, so likely the answer is unknown, unless you want a result of the form "100% of such $n$ can be expressed in such a form" $\endgroup$
    – Stanley Yao Xiao
    Commented Sep 19, 2019 at 11:00
  • 2
    $\begingroup$ #1 can be stated in the form similar to #2: for every odd positive integer $n$, there exists an odd positive integer $m$ such that $\tfrac{n^2+m^2}2$ is prime. (Take $m=a-b$.) $\endgroup$
    – Max Alekseyev
    Commented Sep 19, 2019 at 11:22
  • $\begingroup$ I think #2 is equivalent to "every vertical line in the Gaussian integers contains a Gaussian prime (excluding the axis)" $\endgroup$
    – Joel Moreira
    Commented Sep 19, 2019 at 12:38
  • 2
    $\begingroup$ Those are all questions of whether polynomials of one variable represent any primes, which are extremely difficult (not to mention the further restrictions in 1) and 3)). You may want to check out Bunyakovsky's conjecture, which at least answers 2). $\endgroup$
    – Wojowu
    Commented Sep 19, 2019 at 13:17
  • $\begingroup$ As Wojowu said: 2) is implied by Bunyakovsky's conjecture (en.wikipedia.org/wiki/Bunyakovsky_conjecture). $\endgroup$
    – GH from MO
    Commented Sep 19, 2019 at 18:53

0

Reset to default

You must log in to answer this question.