Top new questions this week:

Surprising identities / equations

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic …

(soft-question) (big-list)

asked by Calvin Lin 60 votes answered by imranfat 56 votes

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or …

(inequality) (exponential-function)

asked by Ashley Montanaro 25 votes answered by Macavity 53 votes

Word origin / meaning of 'kernel' in linear algebra

It may be the dumbest question ever asked on math.SE, but... Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : …

(linear-algebra) (matrices) (soft-question) (terminology) (math-history)

asked by user19906 22 votes answered by triple_sec 17 votes

Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions

Let $f_n(x)$ be recursively defined as $$f_0(x)=1,\ \ \ f_{n+1}(x)=\sqrt{x+f_n(x)},\tag1$$ i.e. $f_n(x)$ contains $n$ radicals and $n$ occurences of $x$: $$f_1(x)=\sqrt{x+1},\ \ \ …

(integral) (integration) (indefinite-integral) (elementary-functions) (radicals)

asked by Vladimir Reshetnikov 17 votes answered by Harry Peter 0 votes

When is $2^n \pm 1$ a perfect power

Is there an easy way of showing that $2^n \pm 1$ is never a perfect power, except for $2^3 + 1 = 3^2 $? I know that Catalan's conjecture (or Mihăilescu's theorem) gives the result directly, but I'm …

(elementary-number-theory)

asked by Calvin Lin 17 votes answered by Calvin Lin 6 votes

Integral $\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$

I encountered this scary integral $$\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$$ where $_3F_2$ is a generalized hypergeometric function …

(calculus) (integral) (logarithms) (closed-form) (hypergeometric-function)

asked by Nik Z. 16 votes answered by Vladimir Reshetnikov 5 votes

Mistaken counterexample to FLT; where's the mistake?

This is taken from the Car Talk puzzler of the week, seen here: http://www.cartalk.com/content/mathematic-mistake-0?question I'll summarize it thusly: A hotshot mathematician calls a press …

(number-theory)

asked by taserian 16 votes answered by marty cohen 22 votes

Greatest hits from previous weeks:

How many triangles

I saw this riddle today, it asks how many triangles are in this picture . I don't know how to solve this (without counting directly), though I guess it has something to do with some recurrence. …

(combinatorics) (puzzle) (recreational-mathematics)

asked by Belgi 22 votes answered by Brian M. Scott 13 votes

How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have …

(vector-analysis)

asked by niko 8 votes answered by carlop 12 votes

Can you answer these?

Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article …

(abstract-algebra) (matrices)

asked by Christiaan Hattingh 5 votes

Question on irreducible representation of a Banach algebra

Let $\cal{A}$ be a Banach algebra, $\cal{X}$ a irreducible left $\cal{A}$-module. If $x,y$ in $\cal{X}$ are linearly independent, there exists an element $a\in\cal{A}$ such that $ax=x$ and $ay=0$. Is …

(abstract-algebra) (representation-theory) (operator-algebras)

asked by Zhonghua Wang 5 votes

Does this sequence always give an integer?

It is known that the $k$-Somos sequences always give integers for $2\le k\le 7$. For example, the $6$-Somos sequence is defined as the following : $$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot …

(sequences-and-series) (number-theory)

asked by mathlove 8 votes