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102

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 ...


67

An example is Pythagorians discovery of irrationality of $\sqrt{2}$. They kept it as a secret for a while because of their special philosophical point of view about the rationality of all numbers in the world. In fact their cosmology were based on a presumption that everything in the nature is made of numbers and their ratios. Some stories say that finally a ...


62

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 ...


61

'The function $f(x)$'. No, the function is $f$. Let $f$ and $g$ be real differentiable functions defined in $\mathbb R$. Some people denote $(f\circ g)'$ by $\dfrac{\mathrm df(g(x))}{\mathrm dx}$. Contrast with the above. I discuss this in greater detail here. The differential equation $y'=x^2y+y^3$. Just a minor variant of 1. Correct would be $y'=fy+y^3$ ...


52

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 ...


48

Actually, I think the best example is of a mathematician who is alive nowadays and proved many results which were unpublished for many years just because he wanted to present them all together in order to solve a famous problem: Fermat's last theorem.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


43

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 ...


41

The inconsistent treatment of raising trig functions to powers: $$ \sin^n x \,.$$ Seriously, starting ab inito $$\sin^2 x$$ could mean either $$\sin( \sin(x) )$$ if you are a quantum mechanic and like to see everything as an operator or as $$(\sin x)^2$$ which is the conventional meaning. So why is $$\sin^{-1} x$$ used for $$\arcsin x$$ (which is vaguely ...


35

Almost all real numbers are transcendental. Almost all real numbers are not constructible. Almost all real numbers are not computable. Almost all real numbers are normal. Almost all real numbers are not definable. Almost all real numbers are not periods.


34

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 ...


29

Double factorial $n!!=n(n-2)(n-4)\cdots$, where the product run through positive integers. At the first time this notation confused me a lot because it looks the same as $(n!)!$ . Similar argument about multifactorial.


27

In the first year at college, I was really confused with the notion of a sequence $$\{ x_n : n \in \mathbb N \}$$ because this could also be a set! Then I discovered $$(x_n)_{n \in \mathbb N}$$ And now I am fine with sequences.


26

We often write $f(n) = O(g(n))$, when in fact $O(g(n))$ is a set, and should be written as $f(n) \in O(g(n))$. Similarly for other asymptotic notation, such as $\Theta$ and $\Omega$.


20

Surely the step from numbers to groups and fields (which is due mostly to 19th-century mathematicians such as Abel, Galois and Dedekind) must count as one one of the greatest leaps forward in history. Much of what was already known about numbers was quickly reproven for abstract algebraic structures – and thus for an infinite number of concrete structures ...


20

In 4 dimensions, it is an open question as to whether there are any exotic smooth structures on the 4-sphere.


19

To modern eyes the definition of metric space by Frechet in 1906 may not seem much, but it paved the way to modern analysis and was a huge leap forward. The ubiquity of metric spaces in virtually all realms of mathematics also shows how profound Frechet's contribution was.


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 ...


17

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. ...


17

Let $[a_0; a_1, a_2, \ldots]$ be the continued fraction expansion of $x$. Then there exists a constant $\kappa\approx 2.685452001$ with the property that, for almost all $x$, $$\lim_{n\to\infty} (a_0a_1\ldots a_n)^{1/n} = \kappa.$$ (Many properties about continued fraction expansions are analogous to properties about the normality of base-$n$ expansions. ...


17

A more or less elementary example I'm quite fond of is the Erdős conjecture on arithmetic progressions, which asserts the following: If for some set $S\subseteq \mathbb{N}$ the sum $$\sum_{s\in S}\frac{1}s$$ diverges, then $S$ contains arbitrarily long arithmetic progressions. I've never seen a heuristic argument one way or the other - I believe ...


16

Don't know whether these count as notational abuses as such, but a few common causes of confusion I have come across are \begin{align} &\log(x)\text{ and }\ln(x)\\ &\sin^2(x)\text{ and }\sin(x)^2\\ &\sin^{-1}(x)\text{ and }\arcsin(x)\\ &\log_2(x)\text{ meaning }\log(\log(x)),\text{ & }\log_2(x)\text{ meaning base }2\\ ...


16

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 ...


15

$a^n+b^n=c^n$ has non-trivial integer solutions if and only if $n\le2$


15

I believe whether or not the Thompson group $F$ is amenable is such question. The paper/article "WHAT is... Thompson's Group" mentions that at a conference devoted to the group there was a poll in which 12 said it was and 12 said it was not. There are in fact papers claiming (at least at the time) to have proofs for both sides. Here are some posts to get an ...


14

My 5 cents in favor of category theory by Samuel Eilenberg and Saunders Mac Lane. Not only did it provide a unifying language for very diverse groups of objects/relations, it was the first (to my very limited knowledge) abstract theory to put the emphasis on morphisms preserving structure (rather than structures themselves), and it paved the way to ...


12

Einstein summation convention is a self-explanatory example. Fourier transforms I feel the majority of people (myself included) abuse notation when describing Fourier transforms. For example, it's common to see: $$\mathcal{F}\{{f(x)}\} = F(\omega)$$ to which my natural response is: uhm, no, I believe you mean $$\mathcal{F_x}\{{f}\} = F$$ or perhaps ...


12

Taken from http://arxiv.org/abs/0902.3961 through this question http://mathoverflow.net/questions/21003/polynomial-bijection-from-mathbb-q-times-mathbb-q-to-mathbb-q . Is $f(x,y)=x^7+3y^7$ injective on $\Bbb Q \times \Bbb Q \ $ ?


11

As a probabilist, I cannot not mention the following, very fascinating to me, story of the life of Döblin and his relationship with the formula (almost) universally known as Itō formula http://en.wikipedia.org/wiki/Wolfgang_D%C3%B6blin


10

Differential cryptanalysis: In 1994, a member of the original IBM DES team, Don Coppersmith, published a paper stating that differential cryptanalysis was known to IBM as early as 1974, and that defending against differential cryptanalysis had been a design goal. According to author Steven Levy, IBM had discovered differential cryptanalysis on its own, ...



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