Problem 791

Solution:
Suppose that $5n+3$ is a prime number. Let $x$ and $y$ be natural numbers such that $x^2 = 2n+1$ and $y^2 = 3n+1$. Then
$$5n+3 = 4(2n+1) - (3n+1) = 4x^2 - y^2 = (2x - y)(2x + y).$$
Since $5n+3$ is a prime number and $x$ and $y$ are natural numbers, we must have $2x - y = 1$, implying that $y = 2x - 1$.

Since $n = y^2 - x^2$, we have that
$$x^2 = 2n + 1 = 2(y^2 - x^2) + 1 = 2((2x - 1)^2 - x^2) + 1 = 6x^2 - 8x + 3.$$
Thus,
$$5x^2 - 8x + 3 = 0$$
and the solutions are $x = 1$ or $x = 3/5$. Since $x^2 = 2n+1$ for a natural number $n$, neither of these $x$ would work and we have a contradiction.