Questions tagged [number-theory]
A field of mathematics studying numbers, their properties and structures that arise from them.
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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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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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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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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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How much did John Nash contribute to proving the Riemann hypothesis?
At quite the end of the movie A Beautiful Mind, John Nash tells a student "I am making progress" (towards proving the Riemann hypothesis (RH)). Actually, how much did Nash contribute to the ...
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Why is the Sophie Germain Identity called thus?
Several authors (z.B.: Arthur Engel in his Problem-Solving Strategies, Alexander Bogomolny in this entry of the Cut the Knot website) refer to the following (straightforward) consequence of 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 ...
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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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$\pi(x)$ 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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How did Mersenne discover Mersenne primes?
So, I was wondering, how did Mersenne come up with the formulae $2^p-1$? Do we have any ideas of how it came to be?
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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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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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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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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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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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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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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 ...
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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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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: $$x^5 + x + 1 = 0$$
...
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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}{\...
Danu♦
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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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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 ...
Danu♦
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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 ...
Danu♦
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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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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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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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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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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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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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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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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 ...
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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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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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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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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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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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