Pythagoras and the Pythagoreans
26
Pierre Fermat (1601-1665), was a court
attorney in Toulouse (France). He was an
avid mathematician and even participated in
the fashion of the day which was to recon-
struct the masterpieces of Greek mathemat-
ics. He generally refused to publish, but
communicated his results by letter.
Are there any other Fermat primes? Here is all that is known to date.
It is not known if any other of the Fermat numbers are prime.
p
2
2
p
+ 1
Factors
Discoverer
0
3
3
ancient
1
5
5
ancient
2
17
17
ancient
3
257
257
ancient
4
65537
65537
ancient
5
4,294,957,297 641, 6,700,417
Euler, 1732
6
21
274177,67280421310721
7
39 digits
composite
8
78 digits
composite
9
617 digits
composite
Lenstra, et.al., 1990
10
709 digits
unknown
11
1409 digits
composite
Brent and Morain, 1988
12-20
composite
By the theorem of Gauss, there are constructions of regular poly-
gons of only 3, 5 ,15 , 257, and 65537 sides, plus multiples,
2
m
p
1
p
2
. . . p
r
sides where the
p
1
, . . . , p
r
are distinct
Fermat primes
.