Thread: Prime number

For any prime . Prove that

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Originally Posted by Kanwar245

For any prime . Prove that

If p= 31, which is NOT divisible by 182: 960/182= 5.274725...

Further p- 1= 30 and p+ 1= 32 and neither divides 182 so I have no idea what you or rgrothee are asserting.

Originally Posted by HallsofIvy

If p= 31, [tex]p^2- 1= 960 which is NOT divisible by 182: 960/182= 5.274725...

Further p- 1= 30 and p+ 1= 32 and neither divides 182 so I have no idea what you or rgrothee are asserting.

My bad, you're right. There's a typo. We need to show that

! Oh, Dear. Well,


Obviously, if p is prime, larger than or equal to 29, then both p+1, and p- 1 are even so it remains to show that one of the factors is divisible 7 and one by 13.

Originally Posted by Kanwar245

For any prime . Prove that

By Fermat’s little theorem, 13 divides . Also by Fermat’s little theorem, 7 divides and hence 7 divides . And obviously 2 divides since is odd. The result follows.

NB: This works for all integers which are coprime with 2, 7, and 13.