Questions tagged [number-theory]

A field of mathematics studying numbers, their properties and structures that arise from them.

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69
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3answers
10k views

What evidence is there that Fermat had a proof for his Last Theorem?

Aside from the fact that Fermat was a genius, is it probable that he actually did have a proof? Some specifics that I think would point one way or another: Would the mathematics of his day allow him ...
59
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1answer
7k views

What's the famous story about a mathematician who gave a talk without saying a word?

Years ago, I read a story about a mathematician who found a numerical counterexample to some conjecture long believed to be true. He gave a talk during which he didn't utter a single word but simply ...
25
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4answers
4k views

Irrationality of the square root of 2

We know that Pythagoreans in Ancient Greece discovered that the square root of two is an irrational number. Why was that discovery historically significant? What value was that knowledge to the ...
23
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4answers
9k views

Ancient Chinese numbering system

It has been said that the invention of zero was a great leap forward, not only in abstract understanding, but in the ability to introduce place value notation and do computations; computing using ...
20
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0answers
556 views

What is the modern significance of Theaetetus's classification of quadratic irrationals?

Before Eudoxus's theory of proportion there was a theory of irrationals based on continued fraction expansions, which Fowler calls anthyphairesis. Theaetetus is said to develop a classification of ...
18
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1answer
3k views

Is the story about Fermat's writing on a margin true?

Is there any evidence that Fermat wrote on the margin of a book "I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." Everyone ...
15
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6answers
2k views

Have numbering systems other than base ten ever been used or popular?

Base ten makes a lot of sense as a numbering system, given the number of digits humans typically have on their hands. That said, some older money systems weren't based on the number of fingers we ...
14
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2answers
3k views

Did ancient Greek mathematicians consider numbers independently of geometry?

I am currently reading Oliver Bryne's edition of Euclid's Elements, and in The Elements many arithmetic propositions are proved geometrically, and it feels to me that numbers are always treated as ...
13
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1answer
648 views

Who first introduced the notation $\mathcal{O}$ in algebraic geometry or algebraic number theory

This is my first question for HSM. If it is consider too specialized for HSM, perhaps it can be migrated to MathOverflow. In algebraic number theory, one frequently denotes the ring of algebraic ...
12
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2answers
579 views

Were transcendental numbers considered rare, pre-Cantor?

Because the real numbers are uncountable and the real algebraic numbers are countable, there are uncountably many transcendental numbers. So there are far more transcendentals than rationals. With the ...
12
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1answer
2k views

What is the origin of the Collatz conjecture?

When was the $3x+1$ problem first proposed? At Wikipedia, I found the following statement (without citation): The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who ...
10
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1answer
10k views

How much contribution did John Nash make to proving Riemann Hypothesis?

At quite the end of the movie "A Beautiful Mind", John Nash tells a student- "I am making progress" (to Riemann hypothesis) Actually, how much contribution he made to the proof of Riemann Hypothesis?
10
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2answers
447 views

What motivated Gauss' quadratic sums?

The proofs of quadratic reciprocity based on quadratic Gaussian sums involve sums of the form $$\sum_{k=1}^{p-1}\left(\frac k p \right)\zeta^{ak}$$ Where $\left(\frac k p \right)$ is a Legendre ...
10
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1answer
364 views

What were the initial applications of finite fields?

Finite fields, I believe, were introduced by Galois in his paper "Sur la théorie des nombres", found on page 398 of his Oeuvres. In this paper Galois introduces the idea of taking a polynomial ...
10
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1answer
389 views

How and where was the notion of a primitive root formulated before Gauss?

Gauss credits Euler (and I think some others) with having known of the existence of primitive roots. How did these predecessors of Gauss formulate the notion of a primitive root without a concept of ...
9
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1answer
702 views

History of Galois Theory after Galois

Galois theory occupies a rather central place in modern number theory, from class field theory, to the Langlands program, to the ideas found in Grothendieck's Esquisse d'un Programme. But the ...
8
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2answers
286 views

Who first identified $\frac{n}{\ln(n)}$ as an approximation of a prime counting function?

Gauss, in his 1849 letter to Encke, mentions that he noticed the primes have a density approx $\frac{1}{\ln(n)}$. In that letter, he also mentions an integral function for approximating the prime ...
8
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2answers
259 views

History of $0 \in \mathbb N$.

There are different conventions around whether $0 \in \mathbb N$. I know that $\mathbb N = \{0,1,2,3, \cdots\}$ is called the Bourbakian definition of $\mathbb N$. But did this convention really ...
8
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2answers
181 views

The first of Fermat's last theorem

A simple question(s): When and by who was Fermat's famous conjecture (proved by Wiles et al.) called his "last theorem"? Was the original motivation the same as is given nowadays, that this would be ...
8
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1answer
145 views

How did Mersenne discover Mersenne primes?

So, I was wondering, how did Mersenne come up with the formulae $2^p-1$? Do we have any ideas of how it came to be?
8
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1answer
555 views

What Is the Origin of the Twin Primes Conjecture?

Is there any written work by Euclid about consecutive primes differing by two? What work was done on the problem from the time of Euclid about 2,300 years ago to the time of Polignac in 1849?
7
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4answers
229 views

How can we account for the independent inventions of place value all using the same direction?

I was looking up some ancient ways of writing numbers, to use as part of a lesson on the history of discovery of new numbers, for some young students. In particular I looked at the notation used by ...
7
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1answer
638 views

Why is Dirichlet's L-function called "L-function"?

In analytic number theory, the function $$ L(s,\chi_m) = \sum_{n=1}^\infty \frac{\chi_m(n)}{n^s}. $$ is called the Dirichlet L-function and has many important uses in the study of prime numbers. In ...
7
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1answer
137 views

Who first proved that only primes of the form $4k+1$ divide odd integers of the form $n^2+1$?

I am writing a paper and I would like to cite the person(s) who proved that only primes of the form $4k+1$ can evenly divide odd integers of the form $n^2+1$? For example, if $n=8$, $n^2 + 1 = 65$ ...
7
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1answer
294 views

Where does the letter S in "$S$-units" and in localization $S^{-1} R$ come from?

In number theory, we may encounter the notion of $S$-unit, $S$-integer, etc. where $S$ is a finite set of prime numbers (for simplicity). For instance, if $S = \{2,3\}$ then the $S$-integers are the ...
7
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1answer
3k views

Was there a very early culture that's number system was 12-based, like ours is 10-based?

There are several uses of 12 in some old systems of measurement. Some of them make sense given current context (There are 12 lunar cycles per year), however some of them seem to be arbitrarily chosen. ...
7
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1answer
1k views

Why is the Sophie Germain Identity called thus?

Several authors (z.B.: Arthur Engel in his Problem-Solving Strategies, Alexander Bogomolny in this entry of the Cut the Knot website) refer to the following (straightforward) consequence of the ...
7
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1answer
544 views

Who named the fugacity, who coined the variable name and did it already relate to complex analysis?

In Riemanns monumental paper, he expresses a prime counting function as an inverse Mellin transform of the log of the function he analytically continued into the complex plane $$\Pi(x) = \frac{1}{2\...
7
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1answer
294 views

What is the name of this numeral system?

In a XVth century french manuscript on arithmetic and astrology, there is a description of a numeral system as follows (it starts here in the manuscript). Numbers between 1 and 9 are depicted by a ...
6
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2answers
2k views

When was the first recorded occurence of irrational and imaginary number usage in number theory?

I saw a letter of Euler to Lagrange congratulating him on his usage of imaginary numbers in the "analysis devoted to rational numbers alone", was that the first known such usage? What was the likely ...
6
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2answers
405 views

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

While researching on Mersenne numbers, I often stumble upon statements of this nature (it is not verbatim): Huldaricus Regius in 1536 proved that $2^{11}-1$ is not prime, providing a factorisation ...
6
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2answers
502 views

Who extended the Euclidean algorithm to derive the Bezout identity?

Euclidean algorithm is an algorithm that produces the greatest common divisor of two integers. It was described by Euclid as early as in 300 BC. On the other hand, the extended Euclidean algorithm ...
6
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1answer
2k views

What is the most ancient civilization that used base-16 (hexadecimal) number system?

What is the first or most ancient civilization to use a base-16, hexadecimal number system?
6
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0answers
134 views

Did Hardy and Ramanujan miscalculate these values?

When I read Dickson's History Of The Theory Of Numbers Vol-2, I found that there seems to be a mistake in the approximation of partition numbers p(200). For this reason, I found the original text ...
5
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3answers
477 views

Who was L. Aubry?

In his magnificent book Number Theory: An approach through history, from Hammurapi to Legendre, André Weil quotes the article Solution de quelques questions d'analyse indéterminée, by L. Aubry (Sphinx-...
5
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2answers
344 views

Who pioneered the study of the sedenions?

I found lots of background information about the discovery of both imaginary and complex numbers, and enough information about the first two types of hypercomplex numbers; quaternions and octonions (...
5
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1answer
288 views

Who influenced Gauss in his abstract approach to mathematics?

I have studied that Gauss was one of the firsts mathematicians to defend this idea, about the Abstract Math and the conception of number, claiming that "What is calculated (in the sense of things ...
5
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2answers
304 views

How could the people of the past be sure that a * b = b * a?

Let me quote from Terence Tao "Analysis 1": Histocially, the realization that numbers could be treated axiomatically is very recent, not much more than a hundred years old. Then, how could the ...
5
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1answer
53 views

When was the first appearance of the abbreviation RSA?

When was the first publication of the abbreviation RSA (Rivest, Sharmir, Adleman) because it does not appear in Martin Gardner’s article of 1977 which is at the following url: https://simson.net/ref/...
5
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1answer
83 views

Where did the terminology "characteristic" of a field originate?

Where did this terminology first appear, and what was the motivation behind using the word "characteristic" to refer to the property that $1 + 1 + ...+1$ a certain number of times gives us the ...
5
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1answer
725 views

Is there a translation of Gauss' work on Gaussian integers?

Gauss introduced the Gaussian integers in an 1832 Latin work named Theoria residuorum biquadraticorum. I believe there is a German translation available. Is there an English, or possibly French ...
5
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1answer
393 views

Why did Euclid define "a unit" instead of "the unit"?

I know Euclid's Definition VII.1 of a unit only from English and German translations: A unit is (that) according to which each existing (thing) is said (to be) one. [translation by Fitzpatrick] ...
5
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1answer
368 views

Injection of Bernoulli numbers into topology

The Bernoulli numbers appear in the Harer-Zagier formula enumerating gluings of polygons, the Kervaire-Milnor formula for the order of homotopy groups for n-spheres, and (with the connection to the ...
5
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1answer
114 views

Plane/composite numbers as lines?

In the Elements of Euclid a plane number (i.e. a composite number), was represented by a line AB. But, being a plane number a multiplication of two numbers (i.e. two lines, in the mind of a ...
5
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0answers
277 views

What is Hensel's lemma a lemma for?

Was Hensel's lemma originally used for proving some other theorem? Or is it meant to be a standalone result? Why is it a "lemma" and not a theorem?
4
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1answer
229 views

What changes in mathematics resulted in the change of the definition of primes and exclusion of 1?

Why 1 is not prime? I read in this article that G.H Hardy explicitly included 1 as a prime in the first 6 editions of "A Course in Pure Mathematics", published between 1908-1933. He updated ...
4
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1answer
228 views

Origin of the "law of quadratic reciprocity"

Today, "reciprocity" is the standard mathematical word used for quadratic reciprocity and its generalizations. I found that the name dates back to no later than 1832, when a paper of Dirichlet (...
4
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1answer
100 views

Why did Gauss use [1] modulus and [2] its ablative case to define n as a positive divisor of a - b?

Any Latinists here please? 1. Why did Gauss choose modulus? How does it relate to $n >1$ where $a - b = kn$ for some integer $k$? And why the Ablative Case? ag.algebraic geometry - The Origin(s) ...
4
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1answer
159 views

Has any large group of people used a base other than 10, 20 and 60 for ordinary purposes?

Wikipedia's list of numeral systems lists only $10,20,60$ as having been used in history. There are about twenty-five sets of symbols there used by different groups of people, but only three different ...
4
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1answer
316 views

Why didn't Euclid try to assign numbers to lengths?

Preliminary note: With "Euclid" I don't mean a person but the mathematicians of the Euclidean period of which Euclid (if he had been one person) was a representative. I imagine that Euclid could have ...