Date of Award
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.
Putnam, Bette Catherine, "On Generalized Mersenne and Fermat Primes" (2009). Honors Theses. 2096.