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 2^{n}, then for any two integers n and m,
the number S = p^{(3m+1)} + q^{(2m+1)} is divisible by 3.
This is a confusing problem for the following reasons.
1. It repeats twice in the problem. The first time, it uses ^ for powers. The second time, it uses superscripts.
2. It says, "For any two integers n and m, the number p^(3m+1)+q^(2m+1) is divisible by 3." However, the number n does not appear in p^(3m+1)+q^(2m+1).
3. It is wrong. For example, take p=5, q=13, m=1. qp=8=2^3, but p^(3m+1)+q^(2m+1)=5^4+13^3=625+2197=2822 is not divisible by 3.

Posted by Math Man
on 20121202 13:46:37 