On The Existence of Prime Numbers in Polynomial Sequences, And Odd Perfect Numbers

Abstract

It is known that certain polynomials of degree one, with integer coefficients, admit in nitely-many primes. In this thesis, we provide an alternative proof of Dirichlets theorem concerning primes in arithmetic progressions, without applying methods involving Dirichlet characters or the Riemann Zeta func- tion. A more general result concerning multiples of primes in short-intervals is also provided. This thesis also considers problems concerning the existence of odd perfect numbers. The main contribution is a good upper-bound on the largest prime divisor of an odd perfect number. In addition, we show how new results concerning odd perfect numbers or k - perfect numbers can be obtained by applying a property of completely-multiplicative functions.