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114

Niccolò “Tartaglia” Fontana invented the first general method to find the roots of an arbitrary cubic equation (based on earlier work by himself and others on how to solve cubics of particular forms), but kept his method secret so as to preserve his advantage in problem-solving competitions with other mathematicians. He divulged the secret to his student ...


72

William Sealy Gosset, while working at Guinness, developed a way of gauging the quality of raw materials with very few samples ($\implies$ less lab work $\implies$ cheaper!). As the story goes, company policy at Guinness forbade its chemists from publishing their findings. Thus Gosset had to publish under the pseudonym "Student". The results of his work are ...


59

Cryptography is often a good source of such instances: At GCHQ, Cocks was told about James H. Ellis' "non-secret encryption" and further that since it had been suggested in the late 1960s, no one had been able to find a way to actually implement the concept. Cocks was intrigued, and developed, in 1973, what has become known as the RSA encryption ...


55

Actually, I think the best example is of a mathematician who is alive nowadays and proved many results which were unpublished for many years just because he wanted to present them all together in order to solve a famous problem: Fermat's last theorem. His name is Andrew Wiles.


50

I think the answer to this question is, unfortunately, a little difficult. As many will point out, Wolfram is beyond egotistical and that fact definitely colors the reception of the book. There is a long list of (mostly negative) reviews here. The negativity reaches its apex in the review by Cosma Shalizi. There are some positive reviews as well, though, ...


44

6accdae13eff7i3l9n4o4qrr4s8t12ux Added later: I posted the "aenrsw" above in cryptic form as a bit of secretive levity, but in case the link gets broken, the character string is an anagram used by Newton in a 1677 letter to Leibniz to lay claim to the fundamental theorem of calculus without revealing it. A second, famous example, from an 1829 letter ...


40

The writing of Stephen Wolfram's book A New Kind of Science involved Wolfram Research employees. Wolfram considers proofs done by these employees to be covered by their NDAs, which is exemplified by Matthew Cook's proof that the cellular automata Rule 110 is Turing complete. That proof was first written in 1994, but Matthew Cook was dissuaded from trying to ...


37

There are now quite a few excellent ones, but most of these are pitched at fairly sophisticated readers-graduate students or professional mathematicans. The thinking according to such textbooks, of course,is that the readers are very far along in thier mathematical training and are ready to use that mathematics to learn physics at a very high level. Whether ...


32

Let $A$, $B$ be two closed subsets of $[0, 1]$, both with measure $\frac{1}{2}$, and suppose $A\cap B = \emptyset$. Then $$m([0, 1]\setminus(A\cup B)) = m([0, 1]) - (m(A) + m(B)) = 1 - \left(\tfrac{1}{2} + \tfrac{1}{2}\right) = 0.$$ But $[0, 1]\setminus(A\cup B)$ is open and the only open set with measure zero is the empty set, so $[0, 1]\setminus(A\cup ...


32

The bad news is that mathematical notation can be regarded as an incredibly dense kind of compression: you take a very subtle and carefully evolved idea and choose some symbol to represent it. The "quick meaning" of a symbol will typically lose all the subtlety. There's a worse problem: typesetting is a pain, so it's pretty common for mathematicians to use ...


27

These are all symbols that are usually learned in high school or introductory math in college. There is a wiki page that gives names and purpose for most symbols: https://en.wikipedia.org/wiki/List_of_mathematical_symbols Wikipedia may not be a reliable source for some topics, but with math it is usually very good for a run down in an unknown or new topic.


24

Just to get a data point, using Maple I took $2000$ random quintics with coefficients pseudo-random numbers from -100 to 100 (but the coefficient of $x^5$ nonzero). $1981$ of these were irreducible (of course the reducible ones are solvable). All $1981$ irreducible quintics were not solvable. EDIT: Quintics with a rational root are solvable, and these are ...


24

One sparkling gem at the intersection of number theory and geometry is Aubry's reflective generation of primitive Pythagorean triples, i.e. coprime naturals $\,(x,y,z)\,$with $\,x^2 + y^2 = z^2.\,$ Dividing by $z^2$ yields $\,(x/z)^2\!+(y/z)^2 = 1,\,$ so each triple corresponds to a rational point $(x/z,\,y/z)$ on the unit circle. Aubry showed that we can ...


22

There are influential posthumous papers by authors who did not publish them. Galois had an immense influence on algebra because of the publication of something he wrote the night before he died in a duel. He gave us the word "group". Thomas Bayes had an influential posthumous paper in which he found the conditional probability distribution of a random ...


21

The words "never" and "anything" are somewhat restrictive (and we know that absolute statements are always wrong). Nowadays, you can't be recognized at all when you're not publishing. Additionally, the term "Genius" is hard to define... One person came to my mind is Grigori Yakovlevich Perelman. He hardly published anything, did not even defend his ...


20

I do not know whether this is appropriate for an answer: the work being secret, I cannot tell what it is (this is not a joke). And I would not be enough of a mathematician to describe it anyway. It concerns Alexander Grothendieck, 86 today, winner of the Fields Medal in 1966. My original information is from the French magazine La Recherche, n°486, April ...


20

We usually denote $n$-tuples in the form $(a_1,\ldots,a_n)$, so for example $(x,y,z)$ is a triple, $(x,y)$ is a pair, somewhat redundantly $(x)$ could be called a one-tuple and $()$ a (in fact: the) zero-tuple.


19

The expression "vector space" is usually reserved to the case when the coefficients form a field. When the coefficients only form a ring (such as a skew field), the corresponding structure is called a module.


19

Onsager announced in 1948 that he and Kaufman had found a proof for the fact that the spontaneous magnetization of the Ising model on the square lattice with couplings $J_1$ and $J_2$ is given by $M = \left(1 - \left[\sinh 2\beta J_1 \sinh 2\beta J_2\right]^{-2}\right)^{\frac{1}{8}}$ But he kept the proof a secret as a challenge to the physics community. ...


18

One example of a theorem with multiple proofs is the Fundamental Theorem of Algebra (All polynomials in $\mathbb{C}[x]$ have the "right number" of roots). One way to prove this is build up enough complex analysis to prove that every bounded entire function is constant. Another way is to build up algebraic topology and use facts about maps from balls and ...


18

Why do we stop at the exponentiation stage in the arithmetic of natural numbers? We don't actually stop there. Knuth's up-arrow notation generalizes the operations up to any level you want and Conway's chained arrow notation even further. Note though that all these generalizations are applicable only for $n\in\mathbb{N}$. Inability to generalize and ...


17

Descriptive Geometry was invented by Gaspard Monge in 1765 (he was eighteen) for military applications. For 15 years (possibly more) it was classified and kept as a military secret. It is a geometric technique to represent 3-dimensional objects through projections on planes. It allows 3D geometric constructions similar to what is usually done in 2D, through ...


16

I'm not sure I understand your question correctly. Are you asking for elementary results proved using much more complicated theorems? If so, here's my favourite example. We will prove that the $n^{th}$ root of $2$ is irrational, for $n>2$ Suppose $\sqrt[n]{2}=\frac{p}{q}$, then $2 = \frac{p^n}{q^n}$, which implies $p^n=q^n+q^n$ This contradicts ...


16

Since no one has mentioned it, I must add V.I Arnold's excellent "Mathematical Methods of Classical Mechanics". As the title suggests, the book focuses on classical mechanics, which is always a good start for a physics education. I'd suggest reading this or other good classical mechanics texts before moving on to various quantum theories, as it's difficult ...


16

There are infinitely many differentiable functions $F\colon [0,\infty)\to\mathbb{R}$ that satisfy $$ F'(x)=F(2x)\qquad\text{and}\qquad F(0)=1. $$ We will follow the argument outlined by @HenningMakholm in the comments. First, consider the substitution $G(x) = F\left(\dfrac{2^x}{\log 2}\right)$. Plugging this into the above equation gives $$ G\,'(x) = 2^x ...


16

We have the identity $$\sum_{n\geq0}\frac{r_{2}\left(n\right)}{\sqrt{n+a}}e^{-2\pi\sqrt{\left(n+a\right)b}}=\frac{1}{\sqrt{\pi}}\sum_{n\geq0}r_{2}\left(n\right)\int_{0}^{\infty}e^{-\left(n+a\right)x-\pi^{2}b/x}\frac{dx}{\sqrt{x}}. \tag{1}$$ In fact ...


16

Possibly Abbott, Understanding Analysis


15

Sub $x=\tanh{u}$, $dx = \operatorname{sech^2}{u} \, du$. Then the integral is $$\int_{-\infty}^{\infty} du \, \frac{\operatorname{sech^2}{u}}{\pi^2+4 u^2} $$ Now, use Parseval. The Fourier transforms of the pieces of the integrand are $$\int_{-\infty}^{\infty} du \, \frac{e^{i u k}}{\pi^2+4 u^2} = \frac14 \frac{\pi}{\pi/2} e^{-\pi |k|/2} $$ ...


14

Approach 1: For the first integral \begin{align} 2\int^1_{0}\frac{{\rm d}x}{\pi^2+\ln^2\left(\frac{1-x}{1+x}\right)} &=-\frac{4}{\pi}\mathrm{Im}\int^1_0\frac{{\rm d}x}{(\ln{x}+\pi i)(1+x)^2}\tag1\\ &=-\frac{4}{\pi}\mathrm{Im}\int^1_{-1}\frac{{\rm d}x}{\ln{x}(1-x)^2}\tag2\\ ...



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