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 theoremstates that ifpis a prime number, then for any integera, the numbera−^{p}ais an integer multiple ofp. 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

(x^{a}– 1)/(x – 1), then either а = p or p º 1 (mod a).

2. Find all injections f:N®Nsuch that:

f(n + m) + f(n – m) = f(n) – f(m) + f(f(m) + n).

3. The sequence a_{n}is defined as

a_{0}is an arbitrary real number, a_{n}_{+1}= ëa_{n}û (a_{n }– ëa_{n}û)

Show that for every a_{0}:

$m ³ 0, "n ³ m, a_{n}_{+}_{2}= a_{n}.

4. Find the lowest cardinality of А ÍNsuch that there are 2017 different partitions B_{i}Í A and A\ B_{i}, where

lcm(B_{i}) = gcd(A\ B_{i}).

5. Let a и b are two integers. The relation a ÍZxZis 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.