Find all positive integers $m, n$, and primes $p ≥ 5$ such that $m(4m^2 + m + 12) = 3(p^n − 1)$
I factorized L.H.S. and then used the fact that L.H.S. must be odd since R.H.S. is odd
Further as the factors are $(m^2+3)(4m+1)$ then $(m^2+3)$ must be odd since $(4m+1)$ is odd. That means $m$ must be even. I then substituted $m=2k$. Equation became $(4k^2+3)(8k+1)=3p^n$. Then I made two cases : Either $3$ divides $(4k^2+3)$ or $3$ divides $(8k+1)$. But then I couldn't go any further.