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2016-03-19, 12:13   #9
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

8,369 Posts

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.

Last fiddled with by science_man_88 on 2016-03-19 at 12:37