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 infinitely 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 function. 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.