# Tagged Questions

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### Why are smooth numbers called “smooth”?

"Adleman refers to integers which factor completely into small primes as “smooth” numbers." (ME Hellman, JM Reyneri. Advances in Cryptology, 1983: citation link.) Does anyone know what is the ...
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### Was Vinogradov's 1937 proof of the three-prime theorem effective?

Was Vinogradov's first proof of the three-prime theorem effective? Reasons for my question: Vinogradov presented his proof in 1937 in a monograph; the English translation by K.F. Roth and A. ...
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### Ramanujan's tau function

Why was Ramanujan interested in the his tau function before the advent of modular forms? The machinery of modular forms used by Mordel to solve the multiplicative property seems out of context until I ...
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### 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 ...
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### History question: Roth's theorem on approximating algebraic numbers…before Roth

Roth's theorem has two universal quantifies, over irrational algebraic numbers $\alpha$ and over real $\epsilon>0$. Of course the theorem asserts in each instance that the inequality ...
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### What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here. After Ramanujan and ...
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### Le Haut Commissariat qui surveille rigoureusement l'alignement de ses Grandes Pyramides

Yesterday I came across the following one-paragraph summary of the history of the Law of Quadratic Reciprocity in Roger Godement's Analyse mathématique, IV, p.313 (perhaps the only treatise on ...
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### Riemann zeta at even integers

I am talking about this in a course I am teaching, and hence am wondering: what are the various derivations of the values of Riemann zeta function at even integers? There are two incredibly cool ...
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### sums of rational squares

It is a well-known fact that if an integer is a sum of two rational squares then it is a sum of two integer squares. For example, Cohen vol. 1 page 314 prop. 5.4.9. Cohen gives a short proof that ...
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### On a remark of Tait on FLT for the exponent 3

This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below: In the ...
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### Two implicit references in Serre's *Groupes de Galois : le cas abélien*

In his exposé at the Galois bicentenary conference, Serre makes two references which are not quite explicit. The first reference occurs (at 22:30 in the video) when he is talking about Dedekind's ...
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### The first complete proof of the Kronecker-Weber theorem

While the Kronecker-Weber theorem —that every finite abelian extension of $\mathbb Q$ is contained in a cyclotomic field— is always attributed to, well, Leopold Kronecker and Heinrich Martin Weber, ...
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### English or French translation of Gauss' “Summatio Quarumdam Serierum Singularium”

I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an ...
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### who fixed the topology on ideles?

I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not ...
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### A gamma function identity

In some of my previous work on mean values of Dirichlet L-functions, I came upon the following identity for the Gamma function: \frac{\Gamma(a) \Gamma(1-a-b)}{\Gamma(1-b)} + ...
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### What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?

This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...
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### Meaning of Kronecker's comment to Lindemann

At the Mactutor history page, it is said that Kronecker remarked to Lindemann: "What good your beautiful proof on [the transcendence of] π? Why investigate such problems, given that irrational ...
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### 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 ...