Fermat numbers and cyclotomy

The talk is devoted to the history of Fermat numbers and their kin, as well as their role in cyclotomy. Fermat conjecture that Fermat numbers $F_n=2^{2^n}+1$ are prime, turned out to be both false and wrong (actually, Fermat never asserted this as a fact, Mersenne did!) Nevertheless, it is a GREAT conjecture, which played crucial role in the development of number theory, and algebra at large. Rebuttal of this conjecture, factorisation of $F_6$, was the first published paper of Leonhard Euler in number theory. Similarly, a claim that the regular 17-gon can be constructed by ruler and compass, also intimately related to Fermat numbers, was the first mathematical paper by Carl Friedrich Gauss, after which he decided to become a mathematician. After a brief introduction I outline the history of factorisation of Fermat numbers, including the recent progress due to the advent of distributed computations. The Gauss—Wanzel theorem asserts that a regular $n$-gon is constructible by ruler and compass, if and only if $n$ is a product of $2^m$ and pair-wise distinct Fermat primes. It was an absolute shock for me to discover that the whole story of cyclotomy according to Klein, presented in all textbooks, is a COMPLETE FAKE, which ignores the contributions by French, Russian and even Prussian mathematicians. Thus, in "Disquisitiones" Gauss didn't prove necessity in the above theorem, this was only done by Pierre Wanzel some 40 years later. Gauss has not given an actual construction of a 17-gon, he computed $cos(2\pi/17)$. The first such geometric construction was published by Egor Andreevich von Pauker. Also, he computed $cos(2\pi/257)$ some 10 years before Friedrich Richelot and Fischer, Johann Hermes constructed the regular 65537-gon in Koenigsberg, and not in Linden or Goettingen, and so on. However, Gauss has established the sufficiency part of the Gauss—Pierpoint theorem, which is seldom mentioned either.
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