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Autistic Assistant of Gauss used to check Primality

The human computer was the computational prodigy Johann Martin Zacharias Dase (June 23, 1824 - September 11, 1861), who was born in Hamburg, Germany and also died there. He reportedly attended school ...
• 6,516
Accepted

Who discovered this closed form formula for the n-th prime number?

The formula is derived in Willans, On Formulae for the nth Prime Number (1964) (Mathematical Gazette vol. 48, no. 366, pp. 413-415), who references Dickson's History of the Theory of Numbers, ch. ...
• 76k
Accepted

$2^{11} - 1$ and the mystery of Huldaricus Regius

Huldalrichus Regius seems to be the same person as Ulrich Regius, otherwise known as Ulrich Rieger, according to the Consortium of European Research Libraries (CERL) entry for him. His book, ...
• 2,555
Accepted

How did Gauss determine the number of primes?

For what it's worth: As Goldshtein writes, “Evidently Gauss considered the tabulation of primes as some sort of pastime and amused himself by compiling extensive tables on how the primes distribute ...
• 961

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 ...
Accepted

Euler's proof of infinite primes first since Euclid?

There is a proof by Goldbach, which appears in a letter he wrote to Euler in 1730 (a few years before Euler published his product formula for the zeta function). It is as follows: if $F_n=2^{2^n}+1$ (...
• 5,777

$2^{11} - 1$ and the mystery of Huldaricus Regius

Hudalrichus Regius (English spelling?) is perhaps best known as the discoverer of the first perfect number since Euclid, who had identified 6, 28, 496 and 8128. Euclid knew that $2^{n-1}(2^n - 1)$ is ...
• 6,859
Accepted

Was Cramér the first to interpret the PNT's $1/\log(x)$ as probability of primes?

A good place to look is Granville’s paper “Harald Cramér and the distribution of primes numbers.” It is on Granville’s website here. He brings in Cramér’s work starting on the bottom of page 19. The ...
• 5,597

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: $$\prod_{p \leq N} \left(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots\right) \geq 1+\frac{1}{2}+ \cdots+\frac{1}{N}$$...
• 1,872

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• 10.3k

Was Cramér the first to interpret the PNT's $1/\log(x)$ as probability of primes?

As Professor K. Conrad mentions, the short answer to this question is NO. Around 1792 Gauß already knew that the "frequency [of the primes] is on the average inversely proportional to the ...
• 1,872

History of primality testing

According to this link for the paper "A Brief History of Factoring and Primality Testing B. C. (Before Computers)", Cataldi created a method for primality testing which was verified by ...

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