# Tag Info

73

You are most likely referring to the 1903 presentation by American mathematician Frank Cole. The original false conjecture was that the 67-th Mersenne number $M_{67}:=2^{67}-1$ is prime, and it goes back to the preface to Mersenne's own Cogitata Physica-Mathematica (1644). However, Cole was already confirming rather than disproving, that $M_{67}$ is ...

72

The other answer is correct. In addition, there is significant evidence that Fermat did not have a proof of the theorem now known as Fermat's Last Theorem. First, we should note that Fermat was not a professional mathematician, only an amateur. He never published any mathematics himself. With just that, it would not seem strange that he did not publish his ...

33

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.ac.uk/library/special_collections/early_books/fermat.htm#:~:text=When%20reviewing%20his%20copy%20of,to%20fit%20in%20the%20margin. (The page with his most ...

29

There is no way that Fermat could have had anything approaching the now commonly-accepted proof. Almost none of the concepts in that proof were known in any form in Fermat's time. Further, Fermat is known for publishing very few of his proofs; almost none survive today, and even in the 1800s there was significant doubt in the mathematical community that he ...

25

I do not agree on some details of the interpretation regarding the discovery of the irrationality of $\sqrt{2}$ as a confutation of the Pythagoreans [...] belief that all numbers could be constructed as the ratio of 2 numbers. My undestanding is that all "archaic" Greek mathematics shared the (implicit) assumption that, given two magnitudes, e.g. two ...

25

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 concept of real number existed. Therefore, mathematics was essentially split into two independent areas: arithmetic and geometry. Diophantus wrote on arithmetic, ...

19

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 Roman numerals was notably clumsy. Zero is not necessary in order to have a place-value system. Nor is zero necessary (or sufficient) in order to develop ...

16

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 pseudoprimes, so there was a great deal of anticipation surrounding the announcement of his 1959 lecture at Columbia University sponsored by the American ...

15

According to this link, Legend has it that Hippasus first discovered the irrationality of $\sqrt{2}$. The second link in fact mentions a legend that held that supporters of Pythagoras murdered Hippasus -- who allegedly discovered the irrationality of $\sqrt{2}$ on a boat in the middle of the sea -- by throwing him overboard immediately after he informed them ...

13

To this day many people use various systems besides the decimal one. I was surprised to read that "Old Babylonians used the system based on 60":-) Do not we all use their system today? Not only for time but for angle measurement? When the French revolutionary government introduced the decimal system as a standard, they also tried to reform the angle and ...

13

Old Babylonians used base 60 for calculations. (This is also where 60 seconds in a minute and 60 minutes in an hour come from.) According to Wikipedia, the main advantage of this was that it made practical calculations rather easy due to the number 60 having many divisors. Their mathematics was generally developed for the time but that would make up for a ...

13

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 self-taught mathematician and he was published in the journal Sphinx-Oedipe, in l’Intermédiaire des mathématiciens, l’Enseignement mathématique, by Gauthier-...

12

These legends do exist, and have for along time. But few if any specialist historians of the subject believe Pythagoreans discovered irrationality of $\sqrt{2}$. See: Pythagoras vs. the idea of Pythagoras It is very hard to judge of Greek mathematics before Euclid, let alone before Plato, as there is so little evidence. The most widely read single ...

12

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 first publication only appears in 1963, and is by Klamkin, he also gives other names it acquired: Syracuse Problem, Hasse’s Algorithm, Kakutani’s Problem and ...

12

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 were published, perhaps stimulated by the appearance of a German translation of GALOIS' works, in 1889, and the reprinting in book form, in 1897, of GALOlS' ...

11

As I understand, people did not think much about comparing the size of infinite sets before Cantor, so for most of them the question you are asking had no meaning, and they did not think about it. Perhaps few people felt intuitively that a countable set is "smaller" than a continuum, but Cantor was the first to state and prove this. Edit. But there were ...

11

The Celts (ie. Iron Age Britons and their descendents) are reputed to have used a base 20 system. This is meant to be the origins of the Yan Tan Tethera counting system used by many upland sheep farmers well into the 19th Century and surviving today in a few places such as Swaledale. See http://en.wikipedia.org/wiki/Yan_tan_tethera for a list of the ...

11

Your guess is right: the notation $\mathfrak o$ goes back to Dedekind. If you get a copy of Dirichlet-Dedekind's Vorlesungen über Zahlentheorie and look in Dedekind's famous XI-th Supplement, which was the first systematic development of algebraic number theory, you'll see $\mathfrak o$ starting in section 170 when Dedekind defines Ordnung (= Order).

11

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 transformation $\phi$ of $N$ in itself such that $N$ appears as chain of an element not contained in $\phi(N)$. We call this element, which we shall denote in what ...

10

The French language still has the residue of a base-twenty system. Their word for 80 is 'quatre-vingts', which literally translates to 'four twenties'.

10

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 of proportions of Eudoxus-Euclid is equivalent to the theory of real numbers. Euclid's book X contains some very complicated theory of some irrational numbers. ...

9

In Elements IX.36 Euclid proves:"If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect". Translation: if $1+2+\dots+2^{n-1}=2^n-1$ is prime then $2^{n-1}(2^n-1)$ is perfect, meaning that it ...

9

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 1791. In [1] of this fragment Gauss states this approximation of the primes counting function, as well as additional conjecture on the asymptotics of k-prime ...

9

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 prefers a definition that does not make an explicit exception for $1$. The prime/irreducible shift in terminology, on the other hand, does reflect changes in ...

8

The fact that $\sqrt{2}$ existed and is irrational was a blow to the ancient Greeks who only believed in numbers that they could calculate to a certain degree of precision whenever required. Or in other words, they were familiar with rational numbers. The fact that others numbers existed would have carried the same sort of feelings in them as and when we ...

8

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 could just as well have chosen another letter. Answers to the MathOverflow question Why are they called L-functions? mainly also assert this, but also ...

8

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 things already counted) are not substances (thinkable objects for themselves), but relations between two objects counted two by two... The mathematician abstracts ...

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