I have been reading about the "odd Goldbach conjecture" which states that:
Every odd integer greater than $7$ can be written as the sum of three odd primes
I have also been reading about the fact that for large $𝑁$ there are considerably more primes than perfect squares numbers.
At first I have tried to run an experiment and see if any odd number can be written as the sum of one perfect square and one prime number, and although most attempts (in the beginning) were true, some others where false.
So since the "odd Goldbach conjecture" involves three different numbers, I have allowed myself to check if any odd number can be written as the sum of $2$ perfect squares and $1$ prime number.
I have manually checked it for the first few hundreds' results, and so far I have found no counter example for any odd positive integers greater than $3$
A prime number (greater than $3$) can be written as $6n+1$ or $6n-1$ and a perfect square number can be written as $4n$ or $4n + 1$.
Also, even though perfect squares are more rare than prime numbers, unlike prime numbers, perfect squares have $1$ at their disposal ($1^1 = 1$), and unfortunately for primes $1$ is not a prime number.
So my question is:
If there are any counter examples or any proofs? and if not what is the likelihood for this to be true?
Edit per comment: I have not considered $0$ as a perfect square.
Here are some examples of the early results:
$5 = 1^2 + 1^2 + 3$
$7 = 2^2 + 1^2 + 2$
$9 = 1^2 + 1^2 + 7$
$11 = 2^2 + 2^2 + 3$
$13 = 2^2 + 2^2 + 5$
$15 = 2^2 + 2^2 + 7$
$17 = 1^2 + 3^2 + 7$
$19 = 2^2 + 2^2 + 11$
$21 = 2^2 + 2^2 + 13$
$23 = 3^2 + 3^2 + 5$
$29 = 3^2 + 3^2 + 11$
$31 = 5^2 + 2^2 + 2$
$33 = 3^2 + 1^2 + 23$
$35 = 3^2 + 3^2 + 17$
$37 = 3^2 + 3^2 + 19$
$39 = 5^2 + 1^2 + 13$
$41 = 1^2 + 3^2 + 31$
$43 = 4^2 + 2^2 + 23$
$45 = 6^2 + 2^2 + 5$
$47 = 6^2 + 2^2 + 7$
$49 = 1^2 + 5^2 + 23$
$51 = 6^2 + 2^2 + 11$
$53 = 6^2 + 2^2 + 13$
$55 = 3^2 + 3^2 + 37$
,,,
Best Answer
In 1923, Hardy and Littlewood [Some problems of “partitio numerorum”, III, Acta Math., 44 (1923), pp. 1-70] conjectured that all sufficiently large integers $n$ can be written in the form $n=p+a^2+b^2$ where $p$ is a prime, and $a,b$ are integers. In 1959–1960, Linnik [Hardy–Littlewood problem on the representation as the sum of a prime and two squares, Dokl. Akad. Nauk SSSR, 124 (1959), pp. 29-30 and An asymptotic formula in an additive problem of Hardy and Littlewood, Dokl. Akad. Nauk SSSR, 24 (1960), pp. 629-706] confirmed this conjecture.
See also Hooley, On the representation of a number as the sum of two squares and a prime.