Intereting Posts

Finding real cubic root of the equation
Given $a\in\mathbb{R}^2\backslash X$ and $v\in\mathbb{R}^2$, $\exists\delta$ such that $t\in[0,\delta) \Rightarrow a+tv\in \mathbb{R}^2\backslash X$.
Why is a summable family at most countable?
Intuition for dense sets. (Real analysis)
Interesting problem of finding surface area of part of a sphere.
How to show that $\gcd(ab,n)=1$?
Estimate a sum of products
Do there exist two singular measures whose convolution is absolutely continuous?
Quadratic reciprocity: Tell if $c$ got quadratic square root mod $p$
Difference between mutually exclusive events and independent events?
partial integration
Why can't $\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}>2$?
Prove that if $A^2x=x$ then $Ax=x$
visualizing functions invariant (or almost) under modular transformation
Polynomial in $n$ variables

This problem is mentioned in this one, but I think it deserves some attention on its own. So here it is:

For any integers $n,m > 0$:

If $2mn(n+m)(n-m)$ divides $(n^2 + m^2 + 1)(n^2 + m^2 – 1)$, then is it true that $n,m$ are a pair of consecutive Pell-numbers, where the Pell-numbers are given recursively by:

- Doubt regarding divisibility of the expression: $1^{101}+2^{101} \cdot \cdot \cdot +2016^{101}$
- How to do well on Math Olympiads
- Studying for the Putnam Exam
- Olympiad inequality $\frac{a}{2a + b} + \frac{b}{2b + c} + \frac{c}{2c + a} \leq 1$.
- Resource for Vieta root jumping
- Perhaps a Pell type equation

$P_0 = 0$

$P_1 = 1$

$P_{n+2} = 2P_{n-1} + P_{n-2}$.

The converse is definitely true.

See Wikipedia article on Pell numbers, and the OEIS page.

Remark:

Notice that this implies the quotient of $(n^2 + m^2 + 1)(n^2 + m^2 – 1)$ by

$2mn(n+m)(n-m)$ is then 2 if $m < n$ and $-2$ if $n < m$. So that in the case when $n,m$ are coprime we have following, if the above is true:

Let $(a,b,c)$ be a primitive Pythagorean triple. If $ab \mid c^2 – 1$, then $2ab = c^2 – 1$. –

- What five odd integers have a sum of $30$?
- How do I show that the sum $(a+\frac12)^n+(b+\frac12)^n$ is an integer for only finitely many $n$?
- Find the least number b for divisibility
- If $w^2 + x^2 + y^2 = z^2$, then $z$ is even if and only if $w$, $x$, and $y$ are even
- 371 = 0x173 (Decimal/hexidecimal palindromes?)
- Given $a,b$, what is the maximum number which can not be formed using $na + mb$?
- Are there infinitely many rational pairs $(a,b)$ which satisfy given equation?
- Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
- Is $2^{1093}-2$ a multiple of $1093^2$?
- Proof strategy - Stirling numbers formula

- Where do Mathematicians Get Inspiration for Pi Formulas?
- How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}} $
- A subspace $X$ is closed iff $X =( X^\perp)^\perp$
- Show that derivative less than 1 implies contraction.
- Analogue of Lebesgue differentiation theorem in Orlicz spaces
- If $f(a)=f(b)=0$, Show that $\int_a^b xf(x)f'(x)dx=-\frac 12\int^2dx$
- Simple linear map question
- Find the integral closure of an integral domain in its field of fractions
- References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$
- A definite integral related to Ahmed's integral
- Why is $\lim_{x \to c}g(f(x)) = g(\lim_{x \to c}f(x))$
- The Matrix Equation $X^{2}=C$
- Convergence in $L^p$ of $f_n$ implies convergence in $L^1$ of $|f_n|^p$ and $f_n^p$
- Prove that curve with zero torsion is planar
- Nilradical strictly smaller than Jacobson radical.