The mathematician Pierre de Fermat (1601-1655) appears several times in this course. In search of a formula for primes, Fermat considered numbers of the form 2j + 1. Below is an applet analogous to the one we used in the previous section when we studied Mersenne numbers. The format of the output is as before: the first column shows the value of j, the second column indicates whether 2 j + 1 is prime, and the third column shows the value of the 2 j + 1. The values of j run from 1 to M.
Research Question 3
Address the three questions given in the introduction of this chapter for numbers of the form 2 j + 1:
1. Under what conditions is an integer of this form always prime?
2. Under what conditions is an integer of this form always composite?
3. Are there infinitely many primes of this form?
6.3.1 Note on Terminology
We would like to refer to numbers of the form 2 j + 1 as Fermat numbers, but this would not fit the standard usage. Unless j satisfies a condition (which you are supposed to discover), the number is definitely composite. When j satisfies this condition, 2 j + 1 is called a Fermat number. In any case, a Fermat prime is a prime of the form 2 j + 1.
Section 6.1 | Section 6.2 | Section 6.3 | Section 6.4
Copyright © 2001 by W. H. Freeman and Company