33 votes
Accepted

Is the story about Fermat's writing on a margin true?

Yes, it is true. Fermat's own copy was used in the publication of Diophantus by Fermat's son Samuel, and he included Fermat's notes. The original with Fermat's handwriting is lost. https://www.joh.cam....
Alexandre Eremenko's user avatar
27 votes
Accepted

Did ancient Greek mathematicians consider numbers independently of geometry?

The answer is yes. There was a split. First of all, for the Greek mathematics (and very long after them) "numbers" were integers. "Rational numbers" were called fractions, and no ...
Alexandre Eremenko's user avatar
17 votes
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How much did John Nash contribute to proving the Riemann hypothesis?

Nash was known to have been captivated by RH at an early age after reading E.T. Bell's Men of Mathematics. He had confided in some friends and colleagues that he had an idea that might work involving ...
nwr's user avatar
  • 6,849
15 votes

Who was L. Aubry?

I am Camille Aubry, granddaughter of Léon Aubry (1882-1947), and I thank you for your interest in my great-grandfather. He was a wine grower, farmer, beekeeper, in Jouy-lès-Reims (51). He was also a ...
Camille Aubry's user avatar
13 votes
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Who was N.M. Stephens who refuted the Stronger Feit-Thompson Conjecture?

I’m Nelson Stephens’ daughter. He was born on 6th May 1941 and was a Professor of Mathematics at the University of London. He passed away on 8th January 2024. Not sure I can be much help on the maths ...
Sophie's user avatar
  • 146
12 votes
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What is the origin of the Collatz conjecture?

Wikipedia does not actually know that it is from 1937, my guess is that they picked a number between 1929 and 1950. According to Lagarias, who compiled an exhaustive bibliography on the problem, the ...
Conifold's user avatar
  • 75.2k
12 votes
Accepted

History of Galois Theory after Galois

I suggest that you reed B. Melvin Kiernan's The Development of Galois Theory from Lagrange to Artin. In particular, it says there that: In the 1890's a few noteworthy expositions of GALOIS Theory ...
José Carlos Santos's user avatar
11 votes
Accepted

History of $0 \in \mathbb N$.

For modern times: Richard Dedekind, Was sind und was sollen die Zahlen? (1888), page 20: §71. Definition. A system $N$ [Ein system $N$] is said to be simply infinite when there exists a similar ...
Mauro ALLEGRANZA's user avatar
11 votes

Who was N.M. Stephens who refuted the Stronger Feit-Thompson Conjecture?

A google search for [Stephens + On the Feit-Thompson Conjecture" led me to an online copy of the paper, and the bottom of the first page of this paper gives Stephens' affiliation at that time as &...
Dave L Renfro's user avatar
10 votes
Accepted

When was the first recorded occurence of irrational and imaginary number usage in number theory?

Irrational numbers were used by the ancient Greeks when they were discovered. The earliest texts did not survive but there are plenty of them in Euclid. Though they are not called numbers. The theory ...
Alexandre Eremenko's user avatar
10 votes
Accepted

What changes in mathematics resulted in the change of the definition of primes and exclusion of 1?

As to the primality of $1$, nothing really evolved. This is an illustration that some things in mathematics are not right or wrong, they are just a matter of taste. And some fraction of the population ...
Conifold's user avatar
  • 75.2k
9 votes

How could the people of the past be sure that $a \times b = b \times a$?

Multiplication, before the invention of modern (axiomatic) algebra, was defined as the operation giving the area of a rectangle with sides of a particular length.1 Commutativity of multiplication ...
Jonathan Cast's user avatar
9 votes
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Who first identified $\frac{n}{\ln(n)}$ as an approximation of a prime counting function?

Gauss also came up with the more discrete $n/\ln (n)$- in volume 10 of his collected works appears a short (5-6 pages) fragment entitled "asymptotic laws of arithmetics", which is dated to the year ...
user2554's user avatar
  • 4,369
9 votes
Accepted

Who discovered this closed form formula for the n-th prime number?

The formula is derived in Willans, On Formulae for the nth Prime Number (1964) (Mathematical Gazette vol. 48, no. 366, pp. 413-415), who references Dickson's History of the Theory of Numbers, ch. ...
Conifold's user avatar
  • 75.2k
9 votes
Accepted

$2^{11} - 1$ and the mystery of Huldaricus Regius

Huldalrichus Regius seems to be the same person as Ulrich Regius, otherwise known as Ulrich Rieger, according to the Consortium of European Research Libraries (CERL) entry for him. His book, ...
kimchi lover's user avatar
  • 2,535
9 votes

Hilbert's problem list did not include Fermat's last theorem. Why?

Hilbert included in his list a much more general Problem 10: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process ...
Alexandre Eremenko's user avatar
8 votes
Accepted

Why is Dirichlet's L-function called "L-function"?

It's the original notation used by Dirichlet. The reason why he chose L, without commenting on the choice, rather than some other letter is not known. Chances are there is not much of a reason, and he ...
quid's user avatar
  • 1,451
8 votes
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Who influenced Gauss in his abstract approach to mathematics?

Gauss in Disquisitiones Arithmeticae (1799) does indeed express something close to what is now called mathematical formalism and structuralism. He writes: "What is calculated (in the sense of ...
Conifold's user avatar
  • 75.2k
8 votes
Accepted

When and where was Legendre's Conjecture first published?

Actually Legendre proved, starting from a false theorem, that between $L$ and $L+2\sqrt{L}$ there is always a prime number. See the second edition of Essai sur la Théorie des Nombres at page 406 (...
user6530's user avatar
  • 3,850
8 votes
Accepted

Number theory: a quote

"It has been estimated that, at the present stage of our knowledge, one could give a 200 semester course on commutative algebra and algebraic geometry without ever repeating himself." These ...
Chris Leary's user avatar
7 votes
Accepted

What is the most ancient civilization that used base-16 (hexadecimal) number system?

The only traditional use of hexadecimals that I know of, and that one is a bit of a stretch, is in Chinese weight units, e.g. one jīn (斤) was equal sixteen liǎng (兩), etc., see Non Base-10 Number ...
Conifold's user avatar
  • 75.2k
7 votes
Accepted

Who was L. Aubry?

L. Aubry was a French mathematician (most likely a high school teacher) who published 56 papers in mathematical journals and 4 books in the period 1894-1933, (according to Zentralblatt database). The ...
Alexandre Eremenko's user avatar
7 votes

Who first proved that only primes of the form $4k+1$ divide odd integers of the form $n^2+1$?

It would seem to be Euler. Dickson, in Ch. XVI of History of the Theory of Numbers, writes the following: "Euler discussed the numbers $a$ for which $a^2+1$ is divisible by a prime $4n+1=r^2+s^2$. ...
Conifold's user avatar
  • 75.2k
7 votes
Accepted

Origin of the "law of quadratic reciprocity"

The term (loi de réciprocité) was introduced by Legendre in his book Essai sur la Théorie des Nombres (1797). It has (p.214) a section titled Théoréme contenant une loi de réciprocité qui existe entre ...
Conifold's user avatar
  • 75.2k
6 votes

What evidence is there that Fermat had a proof for his Last Theorem?

From the evidence that we have, it is most likely that Fermat never even claimed to have a proof of the FLT, see the extensive discussion at Mathoverflow here. Quoting from the accepted answer: Not ...
6 votes
Accepted

Where does the letter S in "$S$-units" and in localization $S^{-1} R$ come from?

As Francois Ziegler suggests in his comment, the notation $S$ and term $S$-unit might go back to Artin and Whaples in their paper about the product formula: "Axiomatic Characterization of Fields by ...
KCd's user avatar
  • 5,322
6 votes

Who pioneered the study of the sedenions?

There are (at least) two different types of numbers called "sedenions". The first ones were introduced by Muses in 1980, who called them $16$-ions, and renamed into "sedenions" by Carmody in 1988. ...
Conifold's user avatar
  • 75.2k
6 votes
Accepted

Euler's proof of infinite primes first since Euclid?

There is a proof by Goldbach, which appears in a letter he wrote to Euler in 1730 (a few years before Euler published his product formula for the zeta function). It is as follows: if $F_n=2^{2^n}+1$ (...
José Carlos Santos's user avatar
6 votes

Simplest of the many proofs the prime harmonic series diverges

The simplest proof is Euler's original proof. It is based on the identity $$\sum_{n=1}^\infty n^{-s}=\prod_{p}\left(1-p^{-s}\right)^{-1}, s>1.$$ This identity is equivalent to existence and ...
Alexandre Eremenko's user avatar

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