2014/027) Solve in natural numbers 2^x – 3^y = 7

x cannot be odd because if x is odd then

2^x mod 3 = -1 so 2^x – 3^y mod 3 = -1 so it cannot be 7 as 7 = 1 mod 3

y cannot be odd as if y is odd 3^y =3 mod 8

so 2^x – 3y = 5 mod 8 for x >= 3

if x = 1 or 2 2^ x < 7 so 2^x – 3^y = 7 not possible

so x and y both are even

say x = 2a and y = 2b

so 2^2a – 3^2b = 7

or (2^a + 3^b)(2^a- 3^b) = 7

so 2^a + 3^b = 7 and 2^a – 3^b = 1

solving these 2 we get a = 2 and b = 1 or x= 4 and y = 2