Honors Theses

Date of Award


Document Type

Undergraduate Thesis



First Advisor

William Staton

Relational Format



The classical Merseniie and Fermat primes are, respectivel3^ primes of the form 2^' - 1 and 2^‘ + 1. The Mersenne primes have been studied since antiquity. It is known that if 2^ - 1 is prime then k is prime. As of September 2008, there are forty-six such primes known. Fermat primes, of the form 2^’ -h 1, seem to be more rare. It is known that if 2^' -I- 1 is prime, then k must be a power of 2. To date only 2“*^ -I- 1, 2^* -f 1, 2^^ -f-1, 2'^^ -t-1, and 2^^ -|-1 are known to be prime. My work involves generalized Mersenne and Fermat primes. Definition: If 6^' — is prime, where a,b, and k are positive integers with a < b and A: > 3, then is a generalized Mersenne prime. I have been able to prove the following analogues to known theorems on Mersenne primes. Theorem: If 6^’ - is a generalized Mersenne prime, then i) = a -t- 1 and ii) k is prime. Theorem: If p is prime and q is a prime divisor of (a -h 1)^ - then q = I (mod p). Using Mathematica, I have found tens of thousands of generalized Mersenne primes. Definition: If is prime, where a, b, and k are positive integers and k > 2. then + b^ is a generalized Fermat prime. Concerning these primes, I have proven the following. Theorem: If a 7^ 1 and -h 6^’ is prime, then k is a power of 2. and a ^ b (mod 2). While there are only five known classical Fermat primes. I have found thousands of generalized Fermat primes. 11 Ill considering whether or not a number is prime, the following is helpful. Theorem: If q is a prime factor of + 6^^, then q = I (mod It is of interest to note that for generalized fermat primes the condition b = a+1 is not necessary as is seen with such examples as 1Q2 + 12 = 101 6^ + 1^ = 1297 5^ + 2^^ = 641. I continue to investigate these and other primes of special forms.

Accessibility Status

Searchable text



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.