Questions tagged [number-theory]

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

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4
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1answer
219 views

Whose 1930 number theory result is used in characterizing perfect 2-error correcting linear codes?

In Error-Correcting Codes: A Mathematical Introduction (Chapman & Hall, 1998), John Baylis wrote (p.109) Moving on to 2-error correcting linear codes, the condition for perfection of linear codes ...
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1answer
76 views

(Where) does Plato define perfect number?

I've read several texts suggesting that Plato defines "perfect number" in his Republic, book VIII 546 b. However, there's no definition as we can see from - for example - this translation: &...
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113 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) ...
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483 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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108 views

How did Fermat come up with his Last Theorem?

It's usually believed that Fermat's claim that he had a proof for the Last Theorem is false, and that it might have been more of a conjecture. Or considering it took many centuries and advanced ...
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2answers
406 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 ...
2
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1answer
159 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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2answers
284 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 ...
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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 ...
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64 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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73 views

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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2answers
118 views

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 ...
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51 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 ...
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1answer
110 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 ...
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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 ...
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 ...
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2answers
800 views

Who was the first to show that this quintic equation has five radical roots?

Despite this is being a very trivial question, I would like to know who was the first (with reference evidence) to show that the following quintic equation has five radical roots: $$x^5 + x + 1 = 0$$ ...
2
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1answer
86 views

What is the history of these prime counting function approximations?

I am reading several sources and there seems to be a lack of clarity, and some contradiction, about the origins of the most recognised prime counting function approximations: $\pi(n) \sim \frac{n}{\...
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2answers
291 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 ...
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1answer
84 views

When and where was Legendre's Conjecture first published?

When and where did Legendre first publish or write about his conjecture that there is a prime between consecutive square numbers? $$n^2 < p < (n+1)^2$$ I have looked through edition 1 and 2 of ...
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 ...
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1answer
113 views

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 ...
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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?
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513 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 ...
4
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1answer
236 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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104 views

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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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 ...
3
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1answer
234 views

What is the basis of the claim that $F_5$ was fully factored in 1732?

The Wikipedia Page on Fermat numbers states that $F_5$ was "fully factored" in 1732. This appears to be the same time that Euler found that any factor of a Fermat number $F_n$ was of the form $$2^{n+...
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106 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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38 views

Timeline for the earliest work on Frobenius problems

If $a, b$ are positive and coprime integers, then the set of linear combinations of $a$ and $b$ with nonnegative coefficients is all integers past $(a - 1)(b - 1)$; i.e. $\{ \lambda_1 a + \lambda_2 b :...
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136 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 ...
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560 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 ...
4
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1answer
322 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 ...
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 ...
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1answer
72 views

What are the direct predecessors of Lagrange's theory of quadratic forms?

I was reading Stillwell's Mathematics and its History, where Lagrange's theory of quadratic forms is synoptically presented, and I was wondering of what are the direct predecessors of the theory. ...
3
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1answer
120 views

Reference for Euler's Introductio in Analysin Infinitorum

In the following answer it has been claimed that "The reference here is not to Euler's 1737 "factorization" of the harmonic series but to 1748 Introductio in Analysin Infinitorum, where the identity ...
3
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1answer
225 views

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 ...
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1answer
730 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 ...
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256 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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1answer
377 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 ...
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1answer
444 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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1answer
114 views

When was the problem of factoring integers explicitly considered, what was the oldest factoring algorithm?

I know Gauss made it clear that he wanted an efficient solution to the problem in 1801. In his Disquisitiones Arithmeticae he calls all factoring algorithms "laborious and prolix", surely a 19th-...
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363 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 (...
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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 ...
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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 ...
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History of Reciprocity Laws

Does anybody know a freely available overview of the history of Reciprocity Laws, especially the Cubic and Biquadratic ones? Wikipedia I know about Franz Lemmermeyer's book "Reciprocity Laws", but ...
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2answers
580 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 ...
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2answers
742 views

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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58 views

Who first distinguished number theory and numerology? [duplicate]

Who first distinguished number theory and numerology?
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1answer
217 views

What is the modern context of Gauss's work on triangles with integer sides and circumradius?

In chapter V of volume 2 of Dickson's "History of the theory of numbers" (p.191-195), which collects results on "rational" triangles (triangles with integer side lengths), apear several results on the ...