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Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

pr.probability polynomials probability-distributions open-problems asked Aug 25, 2019 at 11:15

mathoverflow.net

29 votes

If $n$ is square-free and $n+1 \mid \sigma(n)$, is $n$ a prime?

elementary-number-theory reference-request prime-numbers divisibility conjectures asked Nov 14, 2022 at 13:26

math.stackexchange.com

22 votes

Next math renderer MathJax v3 versus KaTeX?

discussion feature-request mathjax page-rendering asked Nov 23, 2019 at 11:49

meta.stackexchange.com

16 votes

Continued fraction for Apéry's constant conjectured by The Ramanujan Machine

riemann-zeta conjectures continued-fractions asked Apr 20, 2021 at 21:09

math.stackexchange.com

12 votes

Does Diophantine equation $1+n+n^2+\dots+n^k=2m^2$ have a solution for $n,k \geq 2$?

elementary-number-theory diophantine-equations conjectures asked Sep 11, 2021 at 19:03

math.stackexchange.com

12 votes

Nested radical $\sqrt{x+\sqrt{x^2+\cdots\sqrt{x^n+\cdots}}}$

taylor-expansion nested-radicals asked Mar 5, 2016 at 3:49

math.stackexchange.com

11 votes

Determinant of $n\times n$ matrix with parameter

linear-algebra matrices reference-request determinant alternative-proof asked Mar 16, 2016 at 9:46

math.stackexchange.com

11 votes

Can product of distinct $\frac{3m+2}{2m+1}$ be an integer?

elementary-number-theory asked Jun 9, 2018 at 7:48

math.stackexchange.com

10 votes

When is $\lfloor x^n\rfloor$ a perfect square for all integers $n\geq 1$?

elementary-number-theory ceiling-and-floor-functions square-numbers asked Mar 29 at 17:51

math.stackexchange.com

8 votes

Is polynomial $x^{2n-1}-4nx^{2n-2}-2\binom{2n}{3}x^{2n-4}-\dots-2\binom{2n}{3}x^{2}-4n$ irreducible?

polynomials irreducible-polynomials asked Apr 25, 2018 at 19:14

math.stackexchange.com

Top Answers

124

Visually stunning math concepts which are easy to explain

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Prove $4-\sqrt{2}-\sqrt[3]{3}-\sqrt[5]{5} \gt 0$

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Proof that a group is abelian if every square is the identity.

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Solving two-dimensional recurrence relation $a_{i,\ j}\ =\ a_{i,\ j-1}\ +\ a_{i-1,\ j-1}$

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Divisibility property for sequence $a_{n+2}=-2(n-1)(n+3)a_n-(2n+3)a_{n+1}$

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Given $P(x)=x^{4}-4x^{3}+12x^{2}-24x+24,$ then $P(x)=|P(x)|$ for all real $x$

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"Twin binomial coefficient conjecture": There are no binomial coefficients that differ by $2$, excluding $\binom{n}{1}$ and $\binom{n}{n-1}$.

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For what $n$ can we find a degree $\leq n-2$ polynomial such that $P(i) \in \{0 , 1 \}$ for $i \in [n]$, but not all identical.

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Is infinite sequence of irrational numbers digits mathematically observable?

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Consecutive sequences of not cube-free numbers

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On an infinitely large chessboard, in how many paths of length $10$ can a knight take and end up in its original position?

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Solve this equation: $3^{3x} - 3^x = (3x)!$

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I would like to determine if exists a polynomial $R$ with integer coefficients such that $P(x)=Q(R(x))$.

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On the conjecture that $1-\frac 13 + \frac 16 +\frac 1{10} -\frac 1{15}+\cdots = 1\frac 19$

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Find a counterexample: For every antiprime $n>1$, there is a prime divisor $p$ such that $n/p$ is an antiprime

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Methods to see if a polynomial is irreducible

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Proving that $2\sqrt 3+3\sqrt[3] 2-1$ is irrational

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Family of irreducible polynomials

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Determine if $\sum_{n=1}^\infty \frac{n+4^n}{n+6^n}$ converges

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Does $23^n+1=2x^2$ have only two [positive integer] solutions?

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'Obvious' theorems that are actually false

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Find all the functions $f:\mathbb{Z}^+ \to \mathbb{Z}^+$ such that $f(f(x)) = 15x-2f(x)+48$.

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Proposals for polymath projects

mathoverflow.net

In how many $4$-digit numbers the sum of two right digits is equal to the sum of two left digits

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Is $X^8+3X^4-53$ irreducible over $\mathbb{Z}[X]$?

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Prove or disprove that, for any $n \in \mathbb{N_+}$, there exist $a,b \in \mathbb{N_+} $ such that $\frac{a^2+b}{a+b^2}=n.$

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Are there primes $p=47\cdot 2^n+1$?

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Proof: not a perfect square

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If $x+\frac{1}{x}=\frac{1+\sqrt{5}}{2}$ then $x^{2000}+\frac{1}{x^{2000}}= $?

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How can we show that $n < 2^n$ for all natural numbers $n$?

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