Let p and q be two different prime numbers greater than 3.Prove that if their difference is 2^n, then for any two integers n and m,the number S=p^(3m+1)+q^(2m+1) is divisible by 3.

Let p and q be two different prime numbers greater than 3.
Prove that if their difference is 2ⁿ, then for any two integers n and m,
the number S = p^(3m+1) + q^(2m+1) is divisible by 3.

Let n=1; let m=1.

Let p=5, then q=7. 5^(3m+1) =625, 7^(2m+1)=343.

343+625=968, which is 2^3*11^2, not divisible by 3.

Let n=1; let m=1.

Let p=11, then q=13. 11^(3m+1) =14641, 13^(2m+1)=2197.

14641+2197=16838, which is 2*8419, not divisible by 3.

 

Edited on December 2, 2012, 11:31 am

Posted by broll on 2012-12-02 11:23:10