Quote:
Originally Posted by PawnProver44
Is there a prime for every power of 3 just by adding 1 to a single digit. In other words, is there always a prime of the form 3^x+10^y for y ≤ x. (I found this to be false, More generally, is there a prime of the form a^x+b^y for fixed a, x, and b if a and b are coprime. For the example 3^x+10^y, I found this to be true for all x values less than 100:

1) we know the two terms must be coprime so if a and b are coprime it fits our necessary condition however there are conditions on the exponents to stop the number from being composite so a and b being coprime are not sufficient.
in general mod 3 we have the following cases:
a=1 b=0> sum of any powers mod 3 will be 1 mod 3
a=1 b=1> sum of any powers mod 3 will be 2 mod 3
a=1 b=2> sum will differ depending on y if y is odd the sum will be 0 mod 3, if y is even sum will be 2 mod 3.
a=2 b=0 sum will differ depending on x if x is odd the sum will be 2 mod 3, if x is even sum will be 1 mod 3.
a=2 b=1 > sum will differ depending on x if x is odd the sum will be 0 mod 3, if x is even sum will be 2 mod 3.
a=2 b=2 > sum now depends on both x and y:
if x and y are same parity ( both odd or both even) sum will be 2 mod 3.
if x and y are opposite parity the sum will be 0 mod 3.
a=0 b=0 > divisible by 3 and not coprime so that's out
a=0 b=1 > sum of any powers mod 3 will be 1 mod 3
a=0 b=2> sum will differ depending on y if y is odd the sum will be 2 mod 3, if y is even sum will be 1 mod 3.
so this narrows down what x and y can be.