Number Theory Proof
Could anyone assist me with this problem...
"Use mathematical induction to prove the following generalization.
Suppose are integers and p is a prime number. If , then for some ." [Hint: The induction step has two cases.]
I believe I can use this theorem without proof:
Suppose a and b are integers and p is a prime number. if p|ab, then p|a or p|b. This theorem comes from the uniqueness of prime factorization section.
Let be the statement that if , then for some .
Base case: is a trivial statement. is the theorem you have.
Assume to be true, that is, if then for some . It remains to show that also holds, that is, if , then for some .
So if we have that , by your theorem, we have 2 cases: (1) or (2) .
(1) By the inductive hypothesis, we already have some such that , namely .
(2) Then let and we're done.