Questions tagged [number-theory]

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

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Special quadratic reciprocity? $(-3/p)_2$ and $(5/p)_2$ in addition to $(-1/p)_2$ and $(2/p)_2$?

Context: since $\mathbb Z/p^\times$ is cyclic for $p$ prime, $-1$ is a square mod $p$ if and only if $(-1)^{{p-1\over 2}}=1$. A significantly subtler, but still classical, case is determining the ...
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1 vote
1 answer
108 views

Number theory: a quote

I remember reading a remark on number theory that went something like this: "there is plenty of material in this topic for an $n$ semester course without having to repeat oneself". Have you ...
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2 votes
2 answers
183 views

Simplest of the many proofs the prime harmonic series diverges

Over the history of mathematics, some key facts have had multiple and different proofs developed for them. Sometimes these different proofs provide a unique insight or understanding of those facts. ...
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4 votes
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Who first proved that there is no five-digit perfect number?

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. The first five perfect numbers are 6, 28, 496, 8128, and 33550336. Nicomachus ...
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1 answer
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Hilbert's problem list did not include Fermat's last theorem. Why?

Fermat's Last Theorem was open for more than 350 years until Andrew Wiles proved it in 1995. Are there possible (historical or other) reasons why David Hilbert did not include this famous open problem ...
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Fermat et l'équation de Pell

Does any of you happen to own and electronic copy of the paper "Fermat et l'équation de Pell" by A. Weil? If I understand correctly, this paper can be found on pages 413-419 of the third ...
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What is the name of factoring identity in modular arithmetic?

$(mq) \mod (pq) = q \left[ m \mod p \right]$ Is there a name for the above identity? One would think it is the modular multiplicative identity but searching that gives: $ a b \mod p = (a \mod p) (b \...
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4 votes
1 answer
112 views

$\pi(x)$ and $\operatorname{li}(x)$ cross infinitely many times -- due to Littlewood 1914 or Schmidt 1903?

In Montgomery-Vaughan's Multiplicative Number Theory I: Classical Theory, they prove in Theorem 15.3 that $\pi(x)-\operatorname{li}(x) = \Omega_{\pm}(x^{\Theta-\epsilon})$ for every $\epsilon>0$ ...
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Origin of other names of the Chinese remainder theorem [duplicate]

I went to Wikipedia and decided to look at the official Chinese article of the Chinese remainder theorem. Google translating back to English it is roughly called "Han Xin counting soldiers". ...
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4 votes
1 answer
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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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2 votes
1 answer
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(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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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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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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6 votes
2 answers
433 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 ...
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1 vote
2 answers
347 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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2 votes
1 answer
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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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14 votes
2 answers
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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 ...
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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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2 votes
0 answers
80 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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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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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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1 answer
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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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18 votes
1 answer
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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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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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1 vote
1 answer
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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 ...
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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 ...
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1 vote
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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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1 vote
0 answers
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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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6 votes
0 answers
145 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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1 answer
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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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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. ...
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2 votes
1 answer
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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 ...
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3 votes
1 answer
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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 ...
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8 votes
2 answers
309 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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2 votes
1 answer
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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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3 votes
1 answer
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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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3 votes
1 answer
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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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6 votes
2 answers
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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5 votes
2 answers
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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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1 vote
0 answers
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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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-1 votes
1 answer
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Who first distinguished number theory and numerology? [duplicate]

Who first distinguished number theory and numerology?
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3 votes
1 answer
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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 ...
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4 votes
1 answer
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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 (...
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0 answers
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How did Hardy and Littlewood formulate the k-tuple conjecture?

Let $\mathcal{H}_k = (h_1,h_2,\cdots,h_k)$ be an admissible k-tuple. The k-tuple conjecture predicts that the number of primes $(b+h_1,b+h_2,\cdots,b+h_k)\in \mathbb{P}^k$ with $b+h_k \leq x$ is: $$\...
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3 votes
1 answer
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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 ...
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3 votes
2 answers
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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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7 votes
1 answer
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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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0 votes
2 answers
133 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 ...
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7 votes
3 answers
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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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