Find all possible
nonzero integer solutions to this equation:
13 1996 Z
---- + ------- = ----
M2 N2 1997
Prove that no further solution to the above equation is possible.
As Jer noted, it will suffice to focus on positive integer solutions.
I started be clearing fractions. This got me
13*1997*N^2 + 1996*1997*M^2 = M^2*N^2*Z
Multiply by Z and move to one side
(M^2*Z)*(N^2*Z) - 13*1997*N^2*Z - 1996*1997*M^2*Z = 0
Then add (13+1996)*1997 to each side and factor:
(M^2*Z - 13*1997) * (N^2*Z - 1996*1997) = 7^2*41*1997
Then at this point the right side has 12 distinct ordered factorizations, corresponding to possible (M^2*Z, N^2*Z). At this point it gets tedious to list out solution sets. Maybe I will come back and grind out the numbers later.
Edited on January 3, 2025, 7:07 pm
Some analytic progress
