Given that a, b and c are positive integers satisfying the equation:

3^a + 5^b = 7^c + 1.

Derive mathematically the possible remainders when a and c are separately divided by 4.

Consider the problem equality, mod 5

5^b mod 5 = 0

3^a and 7^c mod 5 depend on a and c, mod 4, as follows:

a mod 4     3^a mod 5
---------      ------------
    0                  3
    1                  4
    2                  2
    3                  1

c mod 4     7^c mod 5
---------     -------------
    0                 2
    1                 4
    2                 3
    3                 1

so, the only values (a mod 4, c mod 4) which can satisfy the equality are: (0,0), (1,2), (2,3)

There might be other considerations which limit this further, but I haven't figured them out yet.

 
 

Posted by Steve Herman on 2007-01-26 09:22:44