Problem Determine all possible non negative integer pairs (x, y) satisfying this equation: 2^x – 5^y = 3 My Attempt: If x =0, then 5^y = -2, which is a contradiction. If x =1, then 5^y =-1, which is a contradiction. If x = 2, then 5^y = 1, so that y = 0 If x>=3, then we observe that: 3^y = 3(Mod 8), so that y must be odd Let us substitute y = 2s+1, where s is a positive integer. …….(*) Again, if y – y, then 2^x = 4, giving: x = 2 If y =1, 2^x = 8, giving x = 3 For y>=3, substituting y = 2s+ 1 in terms of (*), we obtain: 2^x - 5^(2s+1) = 3 Or, 2^x = 3 (Mod 5) or, x = 4t+3, where s is a non negative integer. So we have: 2^(4t+3) – 5^(2s+1) = 3 Or, 8*(16^t) - 5*(25^s) = 3 For t =1, we obtain s =1, so that: (x, y) = (7, 3) Hence, so far we have obtained (x,y) = (7, 3); (3,1) and (2, 0) as valid solutions to the problem. **** I am unable to proceed any further, and accordingly, I am looking for a methodology giving any further valid solution(s) or any procedure conclusively proving that no further solutions can exist for the given problem.