急~求數論高手

1. (a) Let p be a prime and gcd(a,p)=1. Use Fermat’s theorem to verify that
x≡a^p-2 b(mod p) is a solution of the linear congruence ax≡b(mod p).
(b)By applying part(a), solve the congruences 2x≡1(mod31),6x≡5(mod11),
and 3x≡17(mod29).

2.Assuming that a and b are integers not divisible by the prime p,establish the
following:
(a) If a^p ≡b^p(mod p),then a≡b(mod p)
(b) If a^p≡b^p(mod p), then a^p≡b^p(mod p^2)

3.Employ Fermat’s theorem to prove that, if p is an odd prime, then
(a)1^p-1 + 2^p-1 + 3p-1 +...+(p-1)^p-1 ≡-1(mod p)
(b)1^p + 2^p+3^p+…+(p-1)^p≡0(mod p)

4.Assume that p and q are distinct odd primes such that p-1 |q-1. If gcd(a,pq)=1,
show that a^q-1 ≡ 1 (mod pq)

5.If p and q are distinct primes, prove that
p^q-1 + q^p-1 ≡ 1(mod pq)

6.Establish the statements below:
(a)If the number Mp=2p-1 is composite, where p is a prime, then Mp is a
pseudoprime.
(b)Every composite number Fn=2^2^n + 1 is a pseudoprime(n=0,1,2,…)

7.Confirm that the following integers are absolute pseudoprimes:
(a)1105=5•13•17
(b)2821=7•13•31
(c)2465=5•17•29