Acta Mathematica et Informatica Universitatis Ostraviensis

1. I n t r o d u c t i o n In 1640, Pierre de Fermat conjectured that all numbers F m = 2 2 ' " + 1 for m = 0, 1, 2 , . . . , (1) are prime, which was later found to be incorrect. The numbers Fm are called Fermat numbers after him. If Fm is prime, we say that it is a Fermat prime. Until the end of the 18th century, Fermat numbers were most likely a mathe­ matical curiosity. The interest in the Fermat numbers dramatically increased when the German mathematician C. F. Gauss quite unexpectedly found (see [3, Sect. VII]) t h a t there exists a Euclidean construction (by ruler and compass) of the regular polygon with n sides if n ~ 2 * F m , F m 2 • ••Fm,, where n > 3, i > 0, j > 0, and Fmi, Fm,2,..., Fmj are distinct Fermat primes (for j = 0 no Fermat primes appear in the above factorization of n). Gauss stated t h a t the converse implication is true as well, but did not prove it. This was proved later in 1837 (see [17]). At present we know that the first five members of sequence (1) are prime and that (see [2]) Fm is composite for 5 < m < 32. The compositeness of F5 was found by Leonhard Euler in 1732. In 1855, Thomas Clausen gave the complete factorization of FQ into two prime factors (see [1, p. 185], 2000 Mathematics Subject Classification: 11A07, 11A15, 11A51, 05C20.