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1500 questions
176
votes
10 answers
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
For all $a, m, n \in \mathbb{Z}^+$,
$$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
Juan Liner
176
votes
4 answers
Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$
I am trying to find a closed form for
$$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$
It seems that the answer is
$$\frac{\pi^2}{12}\left( 1-\sqrt{3}\right)+\log(2) \log \left(1+\sqrt{3}…
Shobhit Bhatnagar
- 11,515
176
votes
7 answers
Induction on Real Numbers
One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far.
Of course you have to change some things in the inductive step, when you want…
Baju
- 1,863
176
votes
26 answers
Software for drawing geometry diagrams
What software do you use to accurately draw geometry diagrams?
Lucky
- 1,149
175
votes
9 answers
Why do people use "it is easy to prove"?
Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the phrase "it is easy to see/prove/verify/..." in the…
oleksii
- 913
175
votes
6 answers
Deleting any digit yields a prime... is there a name for this?
My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood. His mother won't indulge his requests, but I often will. Here is the day he wanted 100:
But today he wanted the prime 719, which I obliged. When…
Fixee
- 11,565
175
votes
11 answers
The median minimizes the sum of absolute deviations (the $ {\ell}_{1} $ norm)
Suppose we have a set $S$ of real numbers. Show that
$$\sum_{s\in S}|s-x| $$
is minimal if $x$ is equal to the median.
This is a sample exam question of one of the exams that I need to take and I don't know how to proceed.
hattenn
- 2,147
175
votes
9 answers
Why is $1^{\infty}$ considered to be an indeterminate form
From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by…
user9413
174
votes
6 answers
What are the differences between rings, groups, and fields?
Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
cobbal
- 2,035
174
votes
19 answers
Mathematical ideas that took long to define rigorously
It often happens in mathematics that the answer to a problem is "known" long before anybody knows how to prove it. (Some examples of contemporary interest are among the Millennium Prize problems: E.g. Yang-Mills existence is widely believed to be…
Yly
- 15,292
174
votes
20 answers
Striking applications of integration by parts
What are your favorite applications of integration by parts?
(The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!)
Thanks for your contributions, in advance!
Jon Bannon
- 3,171
174
votes
3 answers
Is $2048$ the highest power of $2$ with all even digits (base ten)?
I have a friend who turned $32$ recently. She has an obsessive compulsive disdain for odd numbers, so I pointed out that being $32$ was pretty good since not only is it even, it also has no odd factors. That made me realize that $64$ would be an…
Brian Rothstein
- 1,843
173
votes
2 answers
Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger.
Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than $1,$ and show that $n < x < n+1$ for some integer $n$. …
MJD
- 65,394
- 39
- 298
- 580
172
votes
15 answers
Is there a general formula for solving Quartic (Degree $4$) equations?
There is a general formula for solving quadratic equations, namely the Quadratic Formula, or the Sridharacharya Formula:
$$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } $$
For cubic equations of the form $ax^3+bx^2+cx+d=0$, there is a set of three…
John Gietzen
- 3,501
172
votes
8 answers
Lesser-known integration tricks
I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now looking for a list or reference for some lesser-known…
GREStudying12345
- 271