Questions tagged [number-theory]
A field of mathematics studying numbers, their properties and structures that arise from them.
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Source for a story about algebraic number theory?
I once read that Hilbert and Hurwitz (I think it was them) once discussed the theorem about unique factorization of ideals/divisors in algebraic number fields in one of their walks. They, the story ...
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Why Did Riemann Venture into Number Theory for the Riemann Hypothesis?
I’m exploring the history behind the Riemann Hypothesis and I found something interesting. We know that Bernhard Riemann was mainly focused on complex analysis, but he also wrote a very important ...
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Who was Nicolò Paganini that discovered the amicable pair 1184, 1210?
Nicolò Paganini (not the violinist) was a 16 yo Italian schoolboy when he discovered that 1184 and 1210 form a pair of amicable numbers. It is in fact, the 2nd smallest such pair, and it did escape ...
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Where did Lagrange prove the Four Squares Theorem?
I am trying to confirm the initial publication of Lagrange's Four Squares Theorem.
Most of my sources give that it was proved by him in $1770$. However, the generally very good Penguin Dictionary of ...
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Was Euler aware of the general form of the characterization of primes of the form $p=x^2+ny^2$ for arbitrary $n>0$?
If $n$ is a positive integer then there is a monic irreducible polynomial $f(x)$ such that if $p$ is an odd prime not dividing $n$ nor the discriminant of $f$ then
$$
p=x^2+ny^2\iff \left(\frac{-n}{p}\...
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How did Gauss determine the number of primes?
In Brian Conrey's article on Riemann's hypothesis, one reads in the very beginning:
On Christmas Eve 1849 Gauss wrote a letter to his former student Encke
in which he described his thoughts about the ...
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David Hilbert's paper: Substitution of the group of cyclotomic field
A question about a notation in David Hilberts's "Ein neuer Beweis des Kroneckerschen Fundamentalsatzes über Abelsche Zahlkörper" (here a german online available source, not sure if there ...
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History of Bernoulli numbers
I have been trying to understand what is the meaning of Bernoulli numbers, but to my mind it has been obscured behind complicated formulas without much explaination. I presume finding the history ...
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History of prime distribution $\lim_{x\to\infty}\frac{\pi(x)}{x}=0$
Let $\pi(x)$ be the prime counting function. It is well-known that
$$\lim_{x\to\infty}\frac{\pi(x)}{x}=0.$$
My question is who is the first person to prove this result?
Did Eratosthenes prove this ...
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Implications of several unpublished "binomial congruences" of Gauss
In article 23 of his first memoir on biquadratic residues, Gauss gave the first example of a binomial-type congruence, which is apparently a quite "deep" result - many authors relate it to ...
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Who proposed terminating decimals as a major set and why are them important in France?
After looking at some school sources in French, it is common to provide the various number sets in the following order
$$\mathbb{N}\subset \mathbb{Z}\subset\mathbb{D}\subset\mathbb{Q}\subset\mathbb{R}\...
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Finite fields as quotients
Although finite fields are usually introduced as field extensions of fields of prime order, they also arise as quotients of number rings; e.g., $GF(9)$ comes from taking the Gaussian integers mod 3 ...
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Question about Felix Klein's "Development of Mathematics in the 19th Century" [closed]
The original version of these photos is Felix Klein's "Development of Mathematics in the 19th Century"
In second photo
In this book, it says class-field and decompose 2 into $(1+i)$ and $(1-...
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Unramified étale cohomology groups & unramified Milnor-Witt K-groups: any relation to notion of unramifiedness in number theory
Let $X$ be an integral locally noetherian smooth
scheme over base field $k$. Then for every point $x \in X^{(1)}$ of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation
ring. ...
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Teiji Takagi's Fondation of Class Field Theory in Terms of Norms
I have a question about Teiji Takagi's original motivation for his groundbreaking reformulation of class field theory implementing the ideal norms $N_{\mathfrak{m}}(L/K)$ as new "key players"...
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How did Fermat handle Frenicle's challenge to find a perfect number between $10^{20}$ and $10^{22}$?
Even perfect numbers have the form $(2^n - 1)2^{n-1}$ where $2^n - 1$ is a prime number. This restricts the possible values of $n$ to $34, 35, 36, 37$. Since $n$ must be prime, only $37$ has to be ...
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When did modular forms start to get studied via algebraic geometry?
I'm looking, for instance, as to when people started studying modular curves and modular forms as sections of line bundles on them, as opposed to the point of view of modular forms being holomorphic ...
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Why is the claim that $\pi$ contains all natural numbers thought by the general public to be proven, despite it being conjecture currently?
We've all heard the popular claim that the expansion of $\pi$ contains every natural number; however, as we can see from numerous sources, such as Mathematics Stack Exchange and Wikipedia, it is ...
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When was the first time the number 0 came to mean a multiple of ten? [closed]
The earliest number notation I can find is the sexagesimal digit notation of cuneiform. From my understanding of that, in order to make numbers bigger than 60, you would just add the smaller numbers ...
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Background of 'Eisenstein series'
It's my first question.
I am studying elliptic function, Among them, I have a question about the Weierstrass elliptic function.
In the Weierstrass elliptic function, their coefficients are expressed ...
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Influence of Edmund Landau's list of four problems
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers: Goldbach's conjecture, the twin prime conjecture, Legendre's conjecture (that there ...
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The symbol h for class numbers
We use $h_K$ conventionally to denote the class numbers of number fields $K$.
But I have never thought why the letter $h$ was used for it.
Why and who used $h$ for the class number?
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Who came up with the proof of "Bézout's identity" that uses the well-ordering principle?
Let $a$ and $b$ be two integers not both of which are equal to zero. It is an important and well-known fact that $\text{gcd}(a,b)=ax_{0}+by_{0}$ for some integers $x_{0}$ and $y_{0}$. Even though this ...
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Who first proved necessity of Euclid's formula for pythagorean triples?
The following well-known formula for pythagorean triples is commonly called Euclid's formula:
If $a, b, c$ are three natural numbers with $a,c$ odd, $b$ even, $\gcd(a,b,c)=1$ and $a^2+b^2=c^2$, then ...
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When was the study of automorphic (and in particular modular) forms become "adelized"?
The theory of automorphic forms is stated in modern language as a function on the adelic points of some reductive group satisfying certain properties (see for instance, Chapter 7 of An Introduction to ...
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Origin of Problem 6 on the 1988 International Mathematical Olympiad
The recent Numberphile video on the famous Problem 6 of the 1988 IMO (mentioned in a recent answer on this site) got me wondering:
Who came up with this problem in the first place, and how did they ...
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Who was N.M. Stephens who refuted the Stronger Feit-Thompson Conjecture?
The Stronger Feit-Thompson conjecture states that:
There exist no distinct prime numbers $p$ and $q$ such that:
$\dfrac {p^q - 1} {p - 1}$ and $\dfrac {q^p - 1} {q - 1}$
are not coprime.
This was ...
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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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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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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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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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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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$\pi(x)$ 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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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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(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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$2^{11} - 1$ 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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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?
$$\newcommand{\lowerbrack}[2]{%
\raise{-#1}{\left[\raise{#1}{#...
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Confusion on the original article by Lucas
I am currently researching on all primality tests deriving from Lucas' original paper Théorie des Fonctions Numériques Simplement Périodiques, which is of course known for its great deal of confusion. ...
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Did ancient Greek mathematicians consider numbers independently of geometry?
I am currently reading Oliver Bryne's edition of Euclid's Elements, and in The Elements many arithmetic propositions are proved geometrically, and it feels to me that numbers are always treated as ...
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Who first proved that the existence of a Euclidean algorithm implies unique factorization?
In Simachew's "A Survey on Euclidean Number Fields", he said that Gauss used the existence of a Euclidean algorithm in Gaussian integers to prove that it has unique factorization. Also, he ...
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First motivation for extending Riemann Zeta to complex domain?
Euler developed the Euler Product Formula which shows that the Riemann zeta function encodes information about the prime.
$$\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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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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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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