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To my knowledge, Fermat primes, i.e. primes of the form $2^{2^n}+1$ were the first to play a role in a mathematical result, namely in the characterization of constructible regular n-gons. Gauss discovered the constructability of the regular Heptadecagon (http://en.wikipedia.org/wiki/Heptadecagonhttp://en.wikipedia.org/wiki/Heptadecagon) in 1796.
Not much later, in 1805, Sophie Germain proved Fermat's conjecture for primes $p$ for which $2p+1$ is also prime.

My Question is: Have there been any mathematical results, that are not related to the properties of prime numbers and, were found prior to Gauss' discovery of the constructability of the regular Heptadecagon, that were formulated via a proper subset of the prime numbers, that was characterized via a parameterized expression.

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    $\begingroup$ Do primes of the form 4k+1 count? (Fermat and Euler on sums of two squares.) Gerhard "Ask Me About System Design" Paseman, 2014.03.31 $\endgroup$ Mar 31 '14 at 18:52
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    $\begingroup$ We have those of the form $2n+1$, too :-) $\endgroup$ Mar 31 '14 at 22:39
  • $\begingroup$ @GerhardPaseman I would not count primes of the form 4k+1 as an example, because they are related to a theorem about properties of prime numbers, but I admit that I need to edit my question in that respect. $\endgroup$ Apr 1 '14 at 3:42
  • $\begingroup$ @MarianoSuárez-Alvarez What is the related mathematical result that was found prior to Gauss? $\endgroup$ Apr 1 '14 at 3:48
  • $\begingroup$ @ManfredWeis, Euclid had shown the infinity of primes of that form, say! $\endgroup$ Apr 1 '14 at 4:04
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An earlier example than the Fermat primes is the class of primes of the form $2^n-1$, the so-called Mersenne primes. These occur in Euclid's theorem that $2^{n-1}(2^n-1)$ is perfect when $2^n-1$ is prime, and in the complementary theorem of Euler that these are all the even perfect numbers.

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  • $\begingroup$ very nice example; I would be surprised if something earlier would be found $\endgroup$ Apr 1 '14 at 3:50

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