Problems that need a steward
A steward is a mathematician who commits to understand and write up whatever the network proves on a problem. Sign in with GitHub, tick one box, and your name goes on the problem. Have a problem of your own? Propose it →
- erdos-1003Are there infinitely many solutions to $\phi(n) = \phi(n+1)$, where $\phi$ is the Euler totient function?Steward this
- erdos-1065Are there infinitely many primes $p$ such that $p = 2^k * q + 1$ for some prime $q$ and $k ≥ 0$? This is mentioned as B46 in Unsolved Problems in Number Theory by Richard K. GuySteward this
- erdos-172Whether every finite colouring of the natural numbers has arbitrarily large finite sets all of whose distinct-element sums and products share one colour.Steward this
- erdos-212Is there a dense subset of ℝ^2 such that all pairwise distances are rational?Steward this
- erdos-251Is $\sum_{n=1}^\infty \frac{p_n}{2^n}$ irrational? Here $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \dots$).Steward this
- erdos-324Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct?Steward this
- erdos-376Whether infinitely many n have the central binomial coefficient C(2n, n) coprime to 105.Steward this
- erdos-406Whether only finitely many powers of two have base-three expansions using only the digits 0 and 1.Steward this
- erdos-410Let $σ_1(n) = σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$. Is it true that $\lim_{k → ∞} σ_k(n)^{\frac 1 k} = ∞$? This is problem (iii) from Erdos, Granville, Pomerance, Spiro "On the normal behavior of the iterates of some arithmetical functions" (page 169 of the book "Analytic Number Theory", 1990).Steward this
- erdos-412Let $σ_1(n)=σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$. Is it true that, for every $m, n ≥ 2$, there exist some $i, j$ such that $σ_i(m) = σ_j(n)$?Steward this
- erdos-417Let$$V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$$and$$V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$$ Does $\lim V(x)/V'(x)$ exist?Steward this
- erdos-51Is there an infinite set $A \subset \mathbb{N}$ such that for every $a \in A$, there is an integer n such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer, then $\frac{n_a}{a} → \infty$ as $a → ∞$?Steward this
- erdos-52Whether for every ε>0 a finite set of integers has either its sumset or its product set of size at least a constant times |A|^(2−ε).Steward this
- erdos-68Whether the series summing 1/(n!-1) over n at least 2 has an irrational value.Steward this
- erdos-727Let $k ≥ 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many $n$?Steward this
- erdos-850Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?Steward this
- erdos-931Let $k_1 \geq k_2 \geq 3$. Are there only finitely many $n_2\geq n_1 + k_1$ such that $$ \prod_{1\leq i\leq k_1}(n_1 + i)\ \text{and}\ \prod_{1\leq j\leq k_2} (n_2 + j) $$ have the same prime factors?Steward this
- erdos-952Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that $\lvert x_{n+1}-x_n\rvert \ll 1$?Steward this
- euclid-primesFor every natural number n there is a prime greater than n.Steward this
- riemann-hypothesisThe Riemann Hypothesis: all non-trivial zeros of the Riemann zeta function have real part $\frac{1}{2}$. That is, if $\zeta(s) = 0$, $s \neq 1$, and $s$ is not a trivial zero $-2(n+1)$ for some $n \in \mathbb{N}$, then $\operatorname{Re}(s) = \frac{1}{2}$. This is the official Millennium Prize Problem as posed by the Clay Mathematics Institute. This uses the
RiemannHypothesistype from Mathlib, which is defined as∀ (s : ℂ), riemannZeta s = 0 → (¬∃ n : ℕ, s = -2 * (n + 1)) → s ≠ 1 → s.re = 1 / 2.Steward this