I recently was passing some time on Project Euler, when I came across this question. It deals with finding Pentagonal Numbers $P_j$ and $P_k$ such that $P_j+P_k$ and $P_j-P_k$ are also pentagonal numbers and $P_j-P_k$ is minimized. Without loss of generality, I am simply going to say that $j>k$ for the sake of solving the problem.
If you are not familiar with Project Euler, the idea of the site is to provide mathematical problems that can be solved with simple programs. It provides very good exercises in computer science and algorithms.
I noticed that some problems on the site can easily be solved with pen and paper (no "brute force" calculations needed). I feel as though this problem (though perhaps not as easily as some) can be solved by hand, although I keep hitting dead ends.
I have started with what I know: There are multiple (perhaps even infinitely many) solutions $j,k,n_1,n_2$ to the following:
\begin{align} P_j + P_k &= P_{n_1}\\ P_j - P_k &= P_{n_2} \end{align}
I want to relate these variables in a way such that I can find a class of solutions to this problem. Then, from there it should be easy to find the solution such that $P_j-P_k$ is minimized.
Using the relations described above and using the formula for a pentagonal number, I arrive at the following:
\begin{align} 3j^2 - j + 3k^2 - k &= 3n_1^2 - n_1\\ 3j^2 - j - 3k^2 + k &= 3n_2^2 - n_2 \end{align}
From here, I am unable to progress in any beneficial way. It seems that I hit a dead end no matter where I go from here. Any help is appreciated, thanks!
EDIT: I am looking for a solution. I simply do not have the proper background in number theory to elaborate on what these hints may be telling me to do. Nevertheless, I am really intrigued by how I could solve this.
Please provide or help me to come up with a specific way of solving this problem. Thanks!
