Answer to Question #23704 in Combinatorics | Number Theory for Lso

Question

This question is with regards to Fermat Last Theorem and Number Theory.

Let p and θ be primes with θ>p such that p is not a factor of θ−1. As θ is a prime, we know that for any x ∈ Z we can write xθ−1 in the form xθ−1 = kθ+1 for some integer k. Furthermore, since the greatest common divisor of p and θ − 1 is 1, we can write 1 = ap + b(θ − 1) for some integers a and b.
(a) Show that every integer x can be written in the form x = yp + jθ for some integer y and some integer j.
(b) Show that for p a prime, any prime φ that satisfies the conditions of Lemma 4.1 has the property that p is a factor of φ − 1.
(c) Prove the First Case of Fermat’s Last Theorem for the exponents 13, 17 and 19. (That is, show that if there is a nonzero integer solution to xp + yp = zp for p = 13 then 13 is a factor of xyz, and so on.)