## Relations , functions, cardinality, integers... challenging problems

Hi there!
I have a few problems with I am stuck on, so I am posting them here. If you could help, that would be awesome! Thanks in advance!

1. Fermat's little theorem states that if p is a prime number, then for any integer a, the number a pa is an integer multiple of p. In the notation of modular arithmetic, this is expressed as
Using this theorem prove:
Given prime number p, show that if there are positive integer x and prime number a such that p divides
(xa – 1)/(x – 1), then either а = p or p º 1 (mod a).

2. Find all injections f: N ® N such that:
f(n + m) + f(n – m) = f(n) – f(m) + f(f(m) + n).

3. The sequence an is defined as
a0 is an arbitrary real number, an+1 = ëanû (an – ëanû)
Show that for every a0:
\$m ³ 0, "n ³ m, an+2 = an.

4. Find the lowest cardinality of А Í N such that there are 2017 different partitions Bi Í A and A\ Bi, where
lcm(Bi) = gcd(A\ Bi).

5. Let a и b are two integers. The relation a Í ZxZ is defined by
x a y iff аx+by=1.
Find all possible за a and b for which a is nonempty.
For all а and b proof existence or absence of each of following properties: irreflexive, symmetric, antysymmetric and transitive.