# A conjecture involving the sum of the squares of prime factors

$x$ is an integer greater than $1$. Consider the following equation: $$(x+1)^2=x^2+p_1^2+p_2^2+\dots+p_n^2$$ where $$x=p_1^{a_1}p_2^{a_2}\dots{p_n^{a_n}}$$ Find all such numbers $x$ satisfying the above equation.

So far, $6$ is the only known solution for the problem (trial and error). Our efforts at "solving" the general case have been met with failure (if a proper solution is even possible). We have solved the special cases of $1$ and $2$ prime factors, however (confirming the above result).

The conjecture:

For the above question, $6$ is the only valid solution. Prove or disprove the statement.

• That's an odd way to write it...wouldn't $2x+1=p_1^2+\cdots +p_n^2$ be easier to work with? – lulu Mar 7 '18 at 14:14
• Uh, yeah, but I thought stating the question as originally posed would be better. – Mainak Roy Mar 7 '18 at 14:15