- #1
K Sengupta
- 113
- 0
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.
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.