Questions tagged [number-theory]
A field of mathematics studying numbers, their properties and structures that arise from them.
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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}{\...
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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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Who first distinguished number theory and numerology? [duplicate]
Who first distinguished number theory and numerology?
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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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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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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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How can we account for the independent inventions of place value all using the same direction?
I was looking up some ancient ways of writing numbers, to use as part of a lesson on the history of discovery of new numbers, for some young students.
In particular I looked at the notation used by ...
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Gauss's proofs of quadratic reciprocity
Gauss gave 6 published and 2 unpublished proofs of quadratic reciprocity (see, e.g., here). I suspect this was to try to understand the "real reason" quadratic reciprocity holds (though please ...
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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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Was there a very early culture that's number system was 12-based, like ours is 10-based?
There are several uses of 12 in some old systems of measurement. Some of them make sense given current context (There are 12 lunar cycles per year), however some of them seem to be arbitrarily chosen. ...
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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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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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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 ...
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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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The earliest definition of number
What is the earliest known written attempt at the definition of number? Is it Euclid's Elements?
A unit is that by virtue of which each of the things that exist is called one.
A number is a multitude ...
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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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What's the famous story about a mathematician who gave a talk without saying a word?
Years ago, I read a story about a mathematician who found a numerical counterexample to some conjecture long believed to be true. He gave a talk during which he didn't utter a single word but simply ...