# Questions tagged [number-theory]

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

66 questions
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### How long would it have taken Cole to multiply the factors of $M_{67}$ on a blackboard?

The famous anecdote of the 1903 announcement of the factorization of $2^{67}-1$ by Frank Nelson Cole has recently been discussed, for example in light of the announcement of another "twitter-sized" ...
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### 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 $n=8$, $n^2 + 1 = 65$ ...
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### What is the intuition behind Brahmagupta’s rule for multiplying negative numbers?

The rule says: The product (or quotient) of two debts is a fortune What I’m struggling with is what exactly is the product of two debts? What accounting need forces one to multiply debts? How do ...
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### 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-...
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### Irrational numbers math in old Roman age [duplicate]

I know that Hippasus proved that $√2$ is irrational number. My question is how were they doing the mathmatical operations like multiplication for rational numbers like 1.41421356237 I can do ...
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### 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] ...
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### Was 360 considered a magic number, possibly?

The number $360$ as the number of units into which the circle is divided has some nice properties: it has as many divisors as a number of its size can have it's nearly the number of days per year ...
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### 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 ...
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### Did Euclid consider circle segments as another magnitude?

[I adapted the question to reflect what I've learned from Alexandre's answer: that Euclid never talks of lengths and areas but only of line segments and figures (like squares). The question itself ...
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### Pythagorean and the number 17

I was looking for some properties of the prime numbers, and I found some curiosities of the number 17 in that site. One of them is this 17 was called by the Pythagoreans - opposition, obstruction,...
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### 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/...
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### Notation $n=efr$ in algebraic number theory

When $\Bbb Q \subset K$ is a field extension of finite degree and when $p \in \Bbb Z$ is a prime number, the ideal $p O_K$ decomposes uniquely as a product $\prod_{i=1}^r P_i^{e_i}$ of prime ideals of ...
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### 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 ...
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### Translation of Gauss' Disquisitiones Arithmeticae

Out of curiosity I was searching for an English translation of the Disquisitiones Arithmeticae, and I found out that there is indeed one. It was translated in English in 1965 by a certain Arthur A. ...
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### 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 ...
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### When was the square of negative numbers specified?

We know that the rules of relative number where laid down in India (a product of 2 debts is a fortune) and in Europe they were spread by Bombelli, who , again, only mentions the product of two minuses....
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### 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 ...
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### The earliest definition of number

What is the earliest known written attempt at the definition of number? Is it Euclid's Elements? A unit is that by virtue of which each of the things that exist is called one. A number is a ...
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### Gauss's proofs of quadratic reciprocity

Gauss gave 6 published and 2 unpublished proofs of quadratic reciprocity (see, e.g., here). I suspect this was to try to understand the "real reason" quadratic reciprocity holds (though please ...
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### 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 ...
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### 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 ...
240 views

### Who was the first to prove that $\pi$ was a real number? [closed]

Recently, there were many topics in sci.math discussed by so many (mathematicians, logicians, physicians, cranks and anti-cranks,..etc) the old definition of $\pi$ that is still considered valid up to ...
187 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 ...
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### Why were people in sixteenth century interested in solving the cubic?

There were (I think) no number theoretic or relation to science that time, and the only impression I get from reading books is that they did it too one-up other persons on competitions. That's not a ...
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### What results for $1 + 2 + 3 + …$ are known?

Euler obtained the famous result $1+2+3+...= -1/12$. Is this the only result for $\Sigma_{n=0}^{\infty}n$ or have other results been derived, for instance in set theory?
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### 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?
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### Can Numerology be said to be the precursor of Number Theory?

My understanding is that alchemy was a precursor to modern chemistry. Some might say that numerology, similarly, was an earlier form of what is now known as number theory, but I feel like it's a more ...
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### reference need about History of prime number development

Im trying to connect my study to a breif history of prime numbers . Also im interesting in reading history of numbers how they come and how they developed . So can any one suggest for this question a ...
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### 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 ...
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### 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?
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### 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?
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### 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 ...
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### Which is the first reference using the terminology “Chinese Remainder Theorem” for this theorem?

The Chinese Remainder Theorem is one of the fundamental theorems in modular arithmetic. As far as I know, this terminology for the theorem is due to the fact that the Chinese mathematicians were the ...
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### When was the Fermat number $F_{32}$ shown to be composite?

The first few Fermat numbers are $F_{0}=2^{2^0}+1=3$, $F_{1}=2^{2^1}+1=5$, $F_{2}=2^{2^2}+1=17$. I read that $F_5$ through $F_{32}$ have been shown to be composite. And right now $F_{33}$ is the ...
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### Original Statement and Proof of Chebyshev’s $\vartheta(x) \sim x$

I am looking for an online source that gives the original statement and proof of Chebyshev’s asymptotic relation $\vartheta(x)\sim x$ where $\vartheta(x)=\sum_{p\leq x}\log p$. I have tried ...
271 views

### Are there primitive Pythagoras triplets (in integers), being with all the terms as powerful numbers?

I'm searching a trusted historical sources about primitive Pythagoras triplets as being powerful integers (numerical examples), or a notable work of impossibility of such a triples, but couldn't find ...
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### mathematics was recreated on a foundation of number concepts rather than geometrical ones

In Richard Courant and Fritz John's book Introduction to Calculus and Analysis Volume I, says In modern times mathematics was recreated and vastly expanded on a foundation of number ...
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### History of elementary proof of Fermat's last theorem for $x^3 + y^3 = z^3$ [closed]

What was the historical back ground that probably Fermat’s could had known about a much simpler proof than the first (Euler’s elementary proof of Fermat’s last theorem for $n = 3$), at least for the ...
248 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?
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### What is the origin of the Collatz conjecture?

When was the 3x+1 problem first proposed? I found at wikipedia the following statement: The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in ...
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### 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 ...
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### How was Leonardo of Pisa's book liber abbaci distributed?

In 1202 Leonardo of Pisa, later known as Fibonacci, wrote, in his book "liber abbaci", his explanation of a means of expressing numbers, and manipulating them, that gave us arithmetic as we know it ...
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### 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 ...
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### What did Dedekind's The Nature and Meaning of Numbers contribute to the founding of Set Theory?

As best as I can tell Dedekind's paper was published in 1887 already several years after Cantor's flurry of papers on Set Theory between 1879-1883. With this in mind my central questions are: 1) What ...
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### Did the Greeks discover the irrational numbers? [duplicate]

My history of maths lecturer claims the Greeks did not discover the irrational numbers. Is this true?
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### Historical development of Möbius function

Recently I have been familiarized with the concept of 'Möbius function'. As far I know the Möbius function is defined by \mu(n) = \begin{cases} 1 & \small \text{if $n$ is a square-free ...
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### 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 ...
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### 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 ...