JCSBimp's Hexagonal Orthogonal

Let m=1313. The square root of m is between 36 and 37. (m+1)/2 is odd, and so the ceiling root and square should be odd as well. c=37 already is. m is c squared minus 56, between 49 and 64. If we then let c ascend to 39, m is c squared minus 208, between 196 and 225. Ascending c to 41, m is c squared minus 368, between 361 and 400.

Can you determine from these observations the minimum amount c must ascend to make the difference between m and c squared a perfect square itself?

It’s an old, old mathematical code. Crack the code, solve factorization!