$x$ 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.