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Old 2018-02-24, 21:19   #9
carpetpool
 
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"Sam"
Nov 2016

4768 Posts
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Quote:
Originally Posted by science_man_88 View Post
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.
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