Mathematics Weekly Newsletter
Mathematics Weekly Newsletter

Top new questions this week:

Eyebrow-raising pattern of number of primes between terms of the Fibonacci number sequence?

So, $$1,1,2,3,5,8,13,21...$$ Any connection to primes? appears not. However, in between the Fibonacci numbers are how much primes? Let's see: 1 and 1 ZERO 1 and 2 NADA 2 and 3 ZILCH 3 and 5 ZIP ...

(prime-numbers) (fibonacci-numbers) (pattern-recognition)  
asked by HyperLuminal 35 votes
answered by nullUser 35 votes

If the decimal expansion of $a/b$ contains "$7143$" then $b>1250$

I recently stumbled upon this really interesting problem: If we have a fraction $\frac{a}{b}$ where $a,b \in \mathbb{N}$ and we know that the decimal fraction of $\frac{a}{b}$ has the numerical ...

(elementary-number-theory) (fractions)  
asked by Silenttiffy 33 votes
answered by Peter Woolfitt 42 votes

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ...

(abstract-algebra) (group-theory) (finite-groups) (examples-counterexamples)  
asked by Walter 33 votes
answered by Dietrich Burde 30 votes

I think I see mysterious lines inside triangles—how to prove their existence?

Lately I've been fooling around with points inside a triangle and the sum of their distances from all sides. This was when I noticed a weird behaviour: For each point I chose there always seemed to ...

(geometry) (triangle)  
asked by Silenttiffy 29 votes
answered by achille hui 6 votes

Computing irrational numbers

I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy?

(algebra-precalculus) (number-theory) (algorithms) (irrational-numbers)  
asked by user242891 28 votes
answered by Ethan Bolker 20 votes

Axiomatic definition of sin and cos?

I look for possiblity to define sin/cos through algebraic relations without involving power series, integrals, differential equation and geometric intuition. Is it possible to define sin and cos ...

(trigonometry) (axioms)  
asked by gavenkoa 23 votes
answered by Steven Taschuk 20 votes

Can two perfect squares average to a third perfect square?

My question is does there exist a triple of integers, $a<b<c$ such that $b^2 = \frac{a^2+c^2}{2}$ I suspect that the answer to this is no but I have not been able to prove it yet. I realize ...

(elementary-number-theory) (pythagorean-triples)  
asked by Elliot 21 votes
answered by Will Jagy 28 votes

Greatest hits from previous weeks:

Finding an equation for a circle given its center and a point through which it passes

No idea how to do this, I used to have these conic shapes committed to memory but I forget them already. I am supposed to find an equation for the circle that has center $(-1, 4)$ and passes through ...

(algebra-precalculus) (circle)  
asked by user138246 5 votes
answered by David Mitra 10 votes

Finding two numbers given their sum and their product

The other day I was messing around on Gbrainy and I was asked a question "which 2 numbers when added together =16, and when multiplied together =55." I know the X and Y are 5 and 11 but I wanted to ...

asked by Kyle H 3 votes
answered by Joe 7 votes

Can you answer these?

Description of free Lie algebra in Weibel's book

In Exercise 7.3.2 in Weibel's book An Introduction to Homological algebra the following description of the free Lie algebra over some $k$-module $M$ is given (where $k$ is any commutative ring): ...

(category-theory) (lie-algebras)  
asked by Martin Brandenburg 4 votes

Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...

(algebraic-geometry) (differential-geometry) (reference-request) (soft-question) (riemannian-geometry)  
asked by user238194 5 votes

Is there a "ping-pong lemma proof" that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la ...

(group-theory) (reference-request) (group-actions) (geometric-group-theory)  
asked by Paul Plummer 11 votes
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