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Proving that (p)1/n is Irrational when p is Prime and n>1

Problem:
Let p be any prime number, and let p satisfy the equation

xn - p = 0

or, equivalently, x = (p)1/n.

and specify that n > 1.11 Prove that x is an irrational number.

Solution:
The proof that, under this condition, x is irrational will be done indirectly by assuming that x is rational, then showing that this assumption leads to a contradiction.

Let x be rational; i.e., let x = a/b where a and b are integers. Then:

p = xn = an/bn = a rational number.

Since p is prime, then p is an integer. Thus, either:

bn = 1 or bn = am where m < n 12
1. If bn = 1, then p = an, and p has factors other than p and 1,13 violating the assumption that p is prime. Therefore, bn 1.
2. If bn = am, where m < n, then p = am+1… an, and p still has factors other than p and 1,14 violating the assumption that p is prime.

Special case: If m = n - 1, then p = a, and bn = pn - 1, or p = b(n/[n - 1]) = b. But if this latter statement is true, then p = 1, and we violate the assumption that p is any prime.

Since the assumption that x is a rational number leads to contradictions in all possible cases, we must conclude that x is irrational.

11 Notice that, for n = 1, x = p, and p = p/1 = p/ = / , which is a rational number. The proof for irrationality is only valid when n > 1.

12 If n = m, then p = 1, and the assumption that p is any prime is violated.

13 Under this condition, p = a...a (taken n times), and a is a factor [divisor] n times over.

14 That is, a is a factor (n-m) times over.