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Seen on https://legauss.blogspot.com/2012/05/para-rir-ou-para-chorar-parte-13.html Theorem: $5!/2$ is even. Proof: $5!/2$ is the order of the group $A_5$. It is known that $A_5$ is a non-abelian simpl …

It is indeed an open problem, as Misha said. But here is a solution. In E. Golod, Some problems of Burnside type. 1968 Proc. Internat. Congr. Math. (Moscow, 1966) pp. 284-289. Izdat. ”Mir”, Moscow, G …

Here is a heuristic argument that there is nothing to explain: The probability that $p$ divides the sum of the preceding primes is $1/p$. So the expected number of primes less than $10^9$ with this p …

As a physicist "in nature" perhaps I can give a few examples that illustrate how non-analytic functions can appear in physics and counter the idea that physicists do not worry about the justification …

No one with any familiarity with his work can doubt that Siegel was one of the greatest mathematicians of the 20th century. Weil was a decisive, opinionated man -- just the type of person who would h …

Here is a quote from the article by Krzysztof Ciesielski: On Stefan Banach and some of his results. Banach J. Math. Anal. 1 (2007), no. 1, 1–10. There is a curious story how Banach got his Ph …

My understanding of this, which is essentially cobbled together from the various news accounts, is as follows: Let $\pi(x;q,a)$ denote the number of primes less than $x$ congruent to $a\bmod q$, and …

I prefer sound to Terry Tao's light. Listen to my voice through a wall. At each moment in time, you hear not just what I am saying now, but also some reverberation from what I said moments ago. So if …

Exercise 15 of Chapter VI (Section 1) of Katznelson's book "An introduction to Harmonic analysis" gives an example of a continuous $L^1$ function $f$ where both sides of the Poisson summation formula …

True story: When I was about to move to Stony Brook to start my PhD, one of my professors took me aside to tell me "You know, when I was a student Milnor was god, and Morse Theory was the bible." I fo …

As an undergraduate, I learned the Sylow theorems in my algebra classes but could never retain either the statement or proof of these theorems in memory except for short periods of time (and in partic …

I was not going to write anything, as I am a latecomer to this masterful troll question and not many are likely going to scroll all the way down, but Paul Taylor's call for Proof mining and Realizabil …

Philosophically, there is essentially only one way to prove that a number is irrational/transcendental, which is to use the fact that there is no integer between 0 and 1. That is, one assumes that th …

We have $$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} \quad \mbox{for} \ Re(s)>1.$$ Taking the derivative with respect to $s$, we get the following $$- \frac{\zeta'(s)}{\zeta(s)^2} = - …

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