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2 votes

Simplest of the many proofs the prime harmonic series diverges

This is my favorite way to present Euler's proof: By the fundamental theorem of arithmetic: $$\prod_{p \leq N} \left(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots\right) \geq 1+\frac{1}{2}+ \cdots+\frac{1}{N}$$...
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5 votes

Simplest of the many proofs the prime harmonic series diverges

The simplest proof is Euler's original proof. It is based on the identity $$\sum_{n=1}^\infty n^{-s}=\prod_{p}\left(1-p^{-s}\right)^{-1}, s>1.$$ This identity is equivalent to existence and ...
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7 votes

Hilbert's problem list did not include Fermat's last theorem. Why?

Hilbert included in his list a much more general Problem 21: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process ...
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2 votes

Fermat et l'équation de Pell

In this short article he explains the history of the solution of Pell's equation. It was proposed by Fermat in his letter to British mathematicians, and solved by Wallis and Brouncker, jointly, in ...
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