Is Integer Factorization Harder Than RSA Factorization?

Square roots of primes are irrational without unique factorization?

• Does anybody know how to show that the square root of a prime number is irrational WITHOUT using unique factorization (the fundamental theorem of arithmetic)? I have a test covering material of this nature tomorrow. This was a homework problem (except we were allowed to use prime factorization). So I gave the really easy proof by contradiction using unique factorization. The instructor gave me full credit but on my paper wrote that it was possible to do without factorization. He's been stressing doing number theoretic proofs without factorization a lot recently in order for everything to generalize to rings, so I want to make sure I can do this one without factorization, but I'm not seeing the solution. Here are some facts that ARE fair game to use. I won't list them all but just the most important ones: * gcd(a,b) is defined to be the unique positive integer c such that c is a common divisor of and and b and any common divisor of a and b divides c. * gcd(a,b) is the smallest positive element of the ideal aZ + bZ = (gcd(a,b))Z; i.e., gcd(a,b) is the smallest positive linear combination of a and b with integer coefficients. * a prime number is defined as an integer p > 1 such that if a is an integer, then either p divides a or gcd(p,a) = 1.

• Answer:

  Using a somewhat different approach, we can prove a general result: If p and q are positive integers such that p^(1/q) is not an integer (p is not a perfect q-power), then p^(1/q) is irrational. Proof Let P(x) = x^q - p. Then, p^(1/q) is a real root of P. By the rational roots theorem, if the rational r = m/n, m and n ≠ 0 integers with gcd = 1, is a root of P, then, m divides p and n divides 1. Hence |n| = 1, |r| = |m| and r is an integer such that r^q = p and p^(1/q) = r, contradicting the assumption that p^(1/q) is not an integer. It follows P has no rational roots. Since p^(1/q) is indeed a real root of P, then p^(1q) is irrational. Corollary: If p > 0 is a prime and q > 1 is an integer, then p^(1/q) is irrational. If q = 2 we have your conclusion.