State True or False. If $$n$$ is a natural number, $$2n + 1$$ and $$3n + 1$$ are squares then $$5n + 3$$ is not a prime.

Let $$2n + 1 = x^2$$ and $$3n + 1 = y^2$$; then $$5n + 3 = 4(2n + 1) - (3n + 1) = (2x - y)(2x + y)$$. If $$2x - y = 1$$, substituting $$y = 2x - 1$$ gives $$5x^2 - 8x + 3 = 0$$, so $$x = 1$$ and $$n = 0$$, which is not natural. So $$2x - y > 1$$, both factors exceed 1, $$5n + 3$$ is composite, and the statement is True.