18 Show that if p is a positive integer such that both p and p² + 2 are prime, then p= 3.

prove that it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

how would you prove this

(1) Let p be a prime number. Show that for any integer a the greatest common divisor of p and a is either p (if pla) or is 1 (if p does not divide a).

Every positive integer is always divisible by 1 and by itself. That is, for every posi- tive integer n we always have 1|n and n|n. We say an integer p ≥ 2 is a prime number if its only divisors are 1 and p. (By convention, 1 is not a prime number.) (a) List the first ten prime numbers. (b) If a positive integer n is not prime, then it is called a composite number. Show that every positive integer n ≥ 2 can be written as a product of prime numbers. (This is called a prime factorization of n. Notice that some primes may need to be repeated. For example, 180 = 2² × 3² × 5 and the primes 2 and 3 are repeated.) Hint: Consider two separate cases based on whether a given number n is prime or composite. For composite numbers, suppose by strong induction that all smaller integers have a prime factorization.

Prove that there is not a finite number of primes.

Prove that the only perfect number of the form n = p2q where p and q are distinct primes is 28.