# Questions tagged [number-theory]

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

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### $2^{11} - 1$ 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 ...
240 views

### Who discovered this closed form formula for the n-th prime number?

The following is a formula for the $n$-th prime number ($[\,]$ represents the floor function). Who was the first person to discover it? The value of this formula: people have been exploring the ...
136 views

### Confusion on the original article by Lucas

I am currently researching on all primality tests deriving from Lucas' original paper Théorie des Fonctions Numériques Simplement Périodiques, which is of course known for its great deal of confusion. ...
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### 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 ...
61 views

### Who first proved that the existence of a Euclidean algorithm implies unique factorization?

In Simachew's "A Survey on Euclidean Number Fields", he said that Gauss used the existence of a Euclidean algorithm in Gaussian integers to prove that it has unique factorization. Also, he ...
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### First motivation for extending Riemann Zeta to complex domain?

Euler developed the Euler Product Formula which shows that the Riemann zeta function encodes information about the prime. $$\zeta(s)=\sum_{n}\frac{1}{n^{s}}=\prod_{p}(1-\frac{1}{p^{s}})^{-1}$$ Riemann ...
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### Was Cramér the first to interpret the PNT's $1/\log(x)$ as probability of primes?

The Cramér probabilistic model of primes is built on the assumption that the probability of $n$ being prime is $$\Pr(n)=\frac{1}{\log (n)}$$ This is not a big leap from the Prime Number Theorem which ...
46 views

### Any historical work on the distribution of prime gaps?

I am looking to see whether historic mathematicians did any work to explain the slightly unexpected distribution of prime gaps? I would have expected Gauss, who studied lists of primes and proposed a ...
102 views

### Euler's proof of infinite primes first since Euclid?

Q. Is it true that Euler's proof of infinite primes was the first since Euclid's which was from around 300BC? Note: By Euler's proof, I mean the use of his Euler product formula for the zeta function ...
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 ...
217 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 ...
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### Where is First-Order Peano Arithmetic first clearly formulated?

I really should know this, but ... When/where/by whom was first-order Peano Arithmetic first clearly and explicitly formulated in a recognizably modern form (perhaps exact notation apart) -- with the ...
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### How were number symbols derived/shaped up?

This question was sitting on my to do list for sometime. So, as I was reading a book on history of science, I came across of a paragraph where the author attempted to give a historical development ...
98 views

### Does this mathematical result have a specific name?

I am not sure if it's new although it may be an easy consequence of some theorem or lemma.The result is as follows: By choosing a set of numbers between $0$ and $n$(for any $n$) picking each number at ...
94 views

### Why didn't the ancient Greeks consider 1 to be odd?

The Wikipedia page on parity currently says: The ancient Greeks considered 1, the monad, to be neither fully odd nor fully even Why didn't they consider 1 as odd? (I am assuming they already had the ...
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### Where in Gauss's nachlass apears his sketch of a geometric proof for the biquadratic reciprocity law?

According to p.200-202 of the book "Reciprocity Laws: From Euler to Eisenstein" by Franz Lemmermeyer, there are two proofs for the biquadratic reciprocity law in Gauss's nachlass: one is a cyclotomic ...
239 views

### 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 $n=8$, $n^2 + 1 = 65$ ...
116 views

### 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 ...
409 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-...
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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 ...
377 views

### 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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### 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....