1
vote
1answer
157 views

First Parameterized Subset of Primes that was Related to a Mathematical Result

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 ...
0
votes
3answers
564 views

Definition of Prime Numbers [duplicate]

The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I ...
12
votes
2answers
725 views

An Euler-proof that cannot be repaired?

Ed Sandifer writes on Euler's wonderful work on prime numbers: The sum of the series of reciprocals of prime numbers $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$ is infinitely large, and is infinitely ...
15
votes
1answer
1k views

Primes represented by two-variable quadratic polynomials

I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its ...
2
votes
1answer
429 views

Whence the k-tuple conjecture?

What is the source of the $k$-tuple conjecture, that every integer tuple $(k_1,\ldots,k_n)$ either contains all members of a congruence class mod a prime or has infinitely many primes amongst ...
12
votes
2answers
3k views

Did Andre Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis? If so, what are a reference and/or a quotation?

Did Andre Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis? If so, what are a reference and/or a quotation?
2
votes
2answers
498 views

Fermat for polynomials, as used in the AKS (Agrawal-Kayal-Saxena) algorithm

The basis for the deterministic polynomial-time algorithm for primality of Agrawal, Kayal and Saxena is (the degree one version of) the following generalization of Fermat's theorem. Theorem ...
9
votes
4answers
1k views

Who first proved that there are at least n^(1-ε) primes up to n?

It's well-known that Hadamard and de la Vallée-Poussin independently proved the Prime Number Theorem in 1896: that $\pi(n)=n/\log n+o(n/\log n)$. I'm curious as to a weaker result: that for any ...
14
votes
3answers
2k views

Twin Prime Conjecture Reference

I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but ...