How Can We Solve 2^m = 3^n + 5 for Non-Negative m and n?

[PrudensOptimus]

Hello,


2^m = 3^n + 5, given m,n >=0

... Find all possible m,n... Any ideas?

 

[Hurkyl]

Try looking at it in varios moduli?

 

[PrudensOptimus]

what does that mean lol pls explain.

 

[Hurkyl]

For example, if you reduce the equation mod 2^m, and there are no values of n such that 3^n + 5 = 0 (mod 2^m), then you've found an upper limit for m.

I don't know if this will work...

 

[Tide]

Try rewriting your equation as
$$2^m - 2 = 3^n + 3$$
That might provide some insight.

 

[robert Ihnot]

Well, two small solutions are: 2^3 = 3+5; 2^5=3^3+5.

 

[devious_]

Try looking up Stroeker and Tijdeman's theorem (a.k.a. the solution to Pillai's conjecture).