## 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?...it 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)

### 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)

### 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)

### 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)

### 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)

### 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)

### 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)

## 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)

### 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 ...

(algebra-precalculus)

### 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)

### 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)
### 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 ...