I need to prove the following about prime numbers:
a) Let n ∈ Z, n > 1. Prove that if n is not divisible by any prime number less than or equal to √n, then n is a prime number.
b) Let n be a positive integer greater than 1 with the property that whenever n divides a product ab where a, b ∈ Z, then n divides a or n divides b. Prove that n is a prime number.
c) Prove that 2 is the only prime of the form n^3 + 1.
Can anyone help with any of those? Thanks.
1) if n is not a prime number, at least n can be written as n=a b where a and b ∈N, if a=b, a=√n, n divisible by √n, if a≠b,
a and b are not both >√n, find the smaller one, n is divisible by the smaller one who is smaller than √n.
2) if n= a b, where a is a prime number, and b isn't divisible by a. we can get p= a c, q= b d, where c and d are prime numbers and isn't divisible by a or d.
hence n is divisible by pq, but not by p or q
3) n³+1 = (n+1)(n²-n+1) 2 is a special case when n²-n+1=1