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1k views

Irrationality of $e$

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^...
27
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695 views

Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel

There is a remark one can find in various books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") saying Dirichlet figured out a proof of ...
12
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187 views

How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,...
12
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599 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor $...
11
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170 views

To whom is the proof that $A_n$ is simple for $n\geqslant 5$ due, in Rotman's book?

The proof in Rotman's book, Introduction to the Theory of Groups, that $A_n$ is simple consists of the observation that $A_n$ is generated by the $3$-cycles, and hence that if a normal subgroup $H\lhd ...
11
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206 views

Why is the Mazur swindle named so?

Often results or techniques in mathematics are called 'theorems'. Sometimes they are called 'tricks'. In no other context have I seen a result called a 'swindle'. Is there a historical reason for this ...
10
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135 views

Did Landau prove that there is a prime on $\bigl(x,\frac65x\bigr)$?

Was Landau the first to prove that there is a prime on $\bigl(x,\frac65x\bigr)$? In his Handbuch $\!^1$ discussing the limit $$\lim_{n\to\infty} \bigl(\pi\bigl((1+\epsilon)x\bigr)-\pi(x)\bigr)=\...
10
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241 views

How do mathematicians know what is known?

How do mathematicians know that what they are researching has not been already know for 200 years? Obviously if they are researching something that is cutting edge it is not a problem, but if one is ...
10
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199 views

Origin of $\mapsto$ notation

Who invented the brilliant $\mapsto$ notation for describing a function's action on a point, as in $x \mapsto x^2$? This is in some sense a counterpart to Who came up with the arrow notation $x \...
10
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617 views

What is the history of “only if” in mathematics?

A quick search on the use of "only if" returns questions asking about its use and meaning in mathematics, such as here, here and here, revealing confusion in its interpretation and use for some people....
9
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152 views

Who is the mathematician “Jacques” in this anecdote?

Who is the mathematician "Jacques" in this anecdote, which I read on p. 260 of The Mathematical Magpie by Clifton Fadiman, who quotes it from the 1942 memoir The Last Time I Saw Paris by Elliot Paul? ...
9
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853 views

Decoding Gauss' Easter Algorithm

In 1800, Gauss published this algorithm for computing the date of Easter in a given year $year$: $a = year \mod 19$ $b = year \mod 4$ $c = year \mod 7$ $k = \lfloor year/100 \rfloor$ $p ...
9
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279 views

History of a combinatoric problem: exchanging numbers by throwing stones

Another user recently asked a question on the Puzzling stack: Two spies throwing stones into a river. Suitably generalised, it becomes: Two spies (Alice and Bob) need to exchange a message. Each ...
9
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560 views

Ramanujan and sum of four cubes

This is more a question on History than proof itself. About a decade ago, a college professor and a Math coach told us about this beautiful theorem: Every multiple of 6 can be written as a sum of ...
9
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406 views

notation for ramification index and inertial degree

For a prime $Q$ lying over a prime $P$, I have seen the ramification index of $Q$ over $P$ denoted by $e(Q|P)$ and the inertial degree of $Q$ over $P$ by $f(Q|P)$. What is the origin of the ...
8
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180 views

Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
8
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140 views

Who is “R. Drabek”?

The book "Algebra für Einsteiger" bei Bewersdorff (I think the English edition is called "Galois Theory for Beginners") starts with a nice quotation: Math is like love; a simple idea, but it can ...
8
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200 views

Priority of the content of a note by Lebesgue from 1905

I refer to a note by Lebesgue Remarques sur la définition de l'intégrale, Bull.Sci.Math. 29 (1905) 272-275 not very known (see pdf for an exposition in English). It is a pedagogical note containing a ...
7
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198 views

Riesz's 1909 proof of the Riesz Representation Theorem

Frigyes Riesz originally proved the Riesz Representation Theorem on $ C[0,1] $ -- here is his 1909 paper in English (original French). He builds a real valued function $ \text{A} $ on $ [0,1] $ ...
7
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307 views

Historical Question about Schur-Zassenhaus Theorem

I couldn't find any historical information about Schur-Zassenhaus theorem in many books or even papers which mention this theorem. I think, Schur proved that if $G$ is a finite group and if $N$ is ...
7
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197 views

Citation for subset complement result

Let $S = \{s_1, \ldots, s_n\} \subset \{1, \ldots, 2n\}$. Consider two operations on $S$, unfortunately both called complement in different setting: let $A(S) = \{1, \ldots, 2n\} \setminus S$ (set ...
7
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421 views

History of line integral.

I'm looking for some information about how the line integral was discovered, since I've been looking for a long time for this. I found that Riemann could integer discontinuity functions, then Poisson ...
6
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83 views

Largest prime known to ancients

As is well known, Fermat couldn't check the primality of $F_{5} = 2^{2^{5}} + 1$. This raises an interesting question : what was the largest prime number that was known to ancients (particularly ...
6
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121 views

Why is the symbol for the exterior product a meet rather than a join?

I've moved this over to HSM. It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (...
6
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63 views

Proofs of Simplicity of $A_n$

There are different proofs of simplicity of the group $A_n$, and one can get at least two proofs by choosing randomly 10 books of the subject, so I will not go into what are these proofs? Rather, I ...
6
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0answers
87 views

What primes were “pending” at the time of Wiles's proof of FLT?

I would like to know what instances of Fermat's Last Theorem were pending at the time of Wiles's proof. More specifically: what families of irregular primes had been discarded as possible ...
6
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159 views

Is Bourbaki unique?

So my understanding is that a while back a group of mostly French mathematicians, under the pseudonym Bourbaki, wrote a somewhat austerely written series titled "Elements of Mathematic(s)" covering a ...
6
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95 views

What did John Nash publish post-illness?

I've searched for this from time to time and never been able to find a single research paper he published since 1960. Every account of his later work seems to finesse this. The Abel prize page for ...
6
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562 views

2015-related question: why are Lucas-Carmichael numbers named after Lucas?

Summary 2015 is a so called Lucas-Carmichael number. I believe (for reasons that I will explain below) that the 'Carmichael' in the name is a reference to ordinary Carmichael numbers and not to the ...
6
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589 views

Mathematics felt by Srinivasa Ramanujan

At the moment I am reading the book Ramanujan's Papers by B.J. Venkatachala, V. Vinay and C.S. Yogananda; when clarifying some doubt with a professor, he told me that Srinivasa Ramanujan used Galois ...
6
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273 views

How Ramanujan find this formula

I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...
5
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52 views

Understanding a medieval approximation

A medieval text (Maimonides's commentary to chapter 2 of Eruvin in my retranslation from the Hebrew) discusses a rectangle whose area is $5000$ square cubits. It reads in relevant part: … that the ...
5
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117 views

Original proof of Taylor's theorem

There are numerous proofs for Taylor's theorem, but What's the original proof for Taylor's theorem (by Taylor?)? In Wikipedia it says: Taylor's theorem is named after the mathematician Brook ...
5
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0answers
185 views

What is the likely future of Univalent Foundations?

Univalent foundations has been hyped up as the foundation for mathematics for the future in articles such as this one. Now I've given HoTT a brief look, and at least seen that it appears on the face ...
5
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63 views

Why row vectors in stochastic processes?

It seems reasonable to state that column vectors $\mathbf{x}$ are the most frequently seen standard notation, often using $\mathbf{x}^\intercal$ to denote a row vector (transposed column vector). ...
5
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0answers
82 views

math historian who don't belong to academia

Is there examples of math historian who don't belong to academia? Is it possible for professionally non-academician to perform good work in the field of the history of mathematics and publish? Does ...
5
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0answers
3k views

What is the origin of “how the Japanese multiply” / line multiplication?

A few months ago I made a video about a way to multiply numbers using lines (here) and it got really popular. I had heard about this trick before and I wanted to trace its origins. It seems to me to ...
5
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0answers
162 views

Why do Mathematicians use $u$ and $v$ as variables?

I'm sure this has happened to you as well: you are reading some hand-written work, the variables used are $u$ and $v$, and at some point the handwriting becomes unclear and you cannot distinguish the $...
5
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74 views

Reference? filler: IRS, Rhind Papyrus, High-school algebra

I believe something like this was included as a filler in one of the MAA journals many years ago. I am searching for the exact reference (for the filler, or an earlier source). Someone dies, and ...
5
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144 views

The mathematical heritage of Lewis Carroll

Which mathematical results has Lewis Carroll, the author of Alice's Adventures in Wonderland, produced? Wikipedia is very vague with regard to this topic and gives us little more than a matrix ...
5
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0answers
331 views

Bloom of Thymaridas

I'm interested in learning more about the Bloom of Thymaridas, a description of which can be found here. Obviously the mathematics behind the identity is not particularly deep from a modern ...
5
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0answers
705 views

Mac Lane and Eilenberg's motivations for category theory

I'm looking to understand the conceptual process that brought Eilenberg and Mac Lane in developing the basic concepts of category theory. I quote Mac Lane's book "Category theory for working ...
5
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0answers
175 views

History Question - Branch Cut

My professor began discussing branch cuts in class today and mentioned that he did not know the origin of the term. Does anyone know the origin of the term and perhaps a source that talks about it?
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73 views

Great mathematical fusions in math history

Development of the mathematics resembles usually a growing tree - from old branches grow new ones. However sometimes domains of mathematics which were separated for the long time are fused together ...
4
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0answers
70 views

Unclear on why Meissel's approach to counting primes works

I am reading through the Wikipedia article on prime counting. The following is presented to describe Meissel's approach: Let $p_1, p_2, \dots, p_n$ be the first $n$ primes. Let $\Phi(m,n)$ be the ...
4
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30 views

History of graph minor concept

I can't find the correct reference to the first introduction of graph minor. There are plenty of strong results on minors (Kuratowski theorem, well-quasi-ordering by Robertson and Seymour, ...) and ...
4
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0answers
37 views

Reference on the history of ergodic theory

I'm looking for some good books on the history of ergodic theory. I'm a Ph.D student in the field, and I am taking Steven Weinberg's advice to learn about the history of my field: http://math....
4
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0answers
86 views

$\tau$-ists and the History of Radian Measure?

Recently, I have been reading about the $\tau$ vs $\pi$ debate. One of the arguments for $\tau$ was that $1\tau$ radian is the whole circle, thus fractions of $\tau$ correspond to the fractions of the ...
4
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72 views

Who was the first to use right and left ideals in a ring?

I know Emmy Noether defined the terms right and left ideal of a ring and made extensive use of them. However, I am interested in knowing whether someone had already coined the term (in the very ...
4
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104 views

In the mean value theorem, we are guaranteed $c$ such that $f'(c) = (f(b)-f(a))/(b-a)$. Does $c$ have a name?

The Mean Value Theorem says approximately that for differentiable $f$, there is a $c \in (a,b)$ such that $$ f'(c) = \frac{f(b)-f(a)}{b - a}. $$ I presume that the number $f'(c)$ is the mean value. My ...