Quote:
Originally Posted by science_man_88
Triple post
You can also show things like:
2^(8x+4)*3^(16y+8)*5^(16z+8)+1 are always divisible by 17.

What you are doing is taking each of the prime powers mod 17 and multiplying them such that the result is always 1. Then add 1 to get 0:
2^(8x+4) = 2^4 = 1 (mod 17)
3^(16y+8) = 3^8 = 1 (mod 17)
5^(16z+8) = 5^8 = 1 (mod 17)
Add these up and you get (1)+(1)+1 = 1 (mod 17)
Then adding 1, you get (1)+1 = 0 (mod 17)
The congruence holds for any values of x, y, z.
Here is another example:
2^(22x+11)*3^(31y)+1 cannot be prime for any integers x, y.
Now as a quick exercise, show that this is true.