Some elementary number theory problems

Problem 1:

Show that if N is an integer greater than zero, either it is a perfect square, or \sqrt{n} is not a rational number.

Problem 2:

Any integer greater than 1, which is not a power of 2, can be written as the sum of two or more consecutive integers.

Problem 3:

If p is any prime, and n \geq 0, prove by induction on n, prove that if a \equiv b \pmod p, then a^{p^{n}} \equiv b^{p^{n}} \pmod {p^{n+1}}.

Please present detailed perfect proofs ! The motivation behind such questions is also to understand rigour of mathematics.

Nalin Pithwa

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