Dataset Viewer
id
stringlengths 6
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| question
stringlengths 67
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| solved
bool 2
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|---|---|---|
Er55c-1
|
Can the smallest modulus of a covering system be arbitrarily large?
| true |
Er55c-2
|
Is it true that, for any $C > 0$, there are infinitely many $n$ such that
$$
p_{n+1} - p_n > C \frac{\log\log n \,\log\log\log\log n}{(\log\log\log n)^2} \log n \;?
$$
| true |
Er55c-3
|
Let $C \ge 0$. Is there an infinite sequence of $n_i$ such that
$$
\lim_{i \to \infty} \frac{p_{n_i+1} - p_{n_i}}{\log n_i} = C \;?
$$
| false |
Er55c-4
|
Let $d_n = p_{n+1} - p_n$. Are there infinitely many $n$ such that
$$
d_n < d_{n+1} < d_{n+2} \;?
$$
| true |
Er55c-5
|
Let $d_n = p_{n+1} - p_n$. Is it true that the set of $n$ satisfying $d_{n+1} \ge d_n$ has density $\tfrac12$, and similarly for $d_{n+1} \le d_n$, and that there are infinitely many $n$ with $d_{n+1} = d_n$? Moreover, for every $k$, do there exist infinitely many runs of length $k$ with
$$
d_n = d_{n+1} = \cdots = d_{n+k}\;?
$$
| false |
Er55c-6
|
Let $n \ge 1$ and
$$
A = \{a_1 < \cdots < a_{\phi(n)}\} = \{1 \le m < n : \gcd(m,n) = 1\}.
$$
Is it true that
$$
\sum_{1 \le k < \phi(n)} (a_{k+1} - a_k)^2 \ll \frac{n^2}{\phi(n)}\;?
$$
| false |
Er55c-7
|
Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Prove that
$$
\sum_{1 \le n \le N} d_n^2 \ll N (\log N)^2\;.
$$
| false |
Er55c-8
|
For every $c \ge 0$, the density $f(c)$ of integers for which
$$
\frac{p_{n+1} - p_n}{\log n} < c
$$
exists and is a continuous function of $c$.
| false |
Er55c-9
|
Let $N_k = 2 \cdot 3 \cdots p_k$ and let
$$
\{a_1 < a_2 < \cdots < a_{\phi(N_k)}\}
$$
be the integers $< N_k$ which are relatively prime to $N_k$. Prove that for any $c \ge 0$, the limit
$$
\frac{\#\bigl\{\,i : a_i - a_{i-1} \le c\,\frac{N_k}{\phi(N_k)},\;2 \le i \le \phi(N_k)\bigr\}}{\phi(N_k)}
$$
exists and is a continuous function of $c$.
| true |
Er55c-10
|
Let $f(n)$ count the number of solutions to $n = p + 2^k$ for prime $p$ and $k \ge 0$. Is it true that
$$
f(n) = o(\log n)\;?
$$
| false |
Er55c-11
|
Let $A \subseteq \mathbb{N}$ satisfy $\lvert A \cap \{1,\dots,N\}\rvert \gg \log N$ for all large $N$. Let $f(n)$ count the number of solutions to $n = p + a$ for $p$ prime and $a \in A$. Is it true that
$$
\limsup_{n\to\infty} f(n) = \infty\;?
$$
| true |
Er55c-12
|
Let $c_1, c_2 > 0$. Is it true that for any sufficiently large $x$, there exist more than $c_1 \log x$ consecutive primes $\le x$ such that the difference between any two is $> c_2$?
| false |
Er56-1
|
If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert = n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$, then
$$
N \gg 2^n.
$$
| false |
Er56-2
|
If $A\subseteq \mathbb{N}$ is such that $A + A$ contains all but finitely many integers, then
$$
\limsup_{n\to\infty} (1_A * 1_A)(n) = \infty.
$$
| false |
Er56-3
|
Given any infinite set $A\subseteq \mathbb{N}$, there is a set $B$ of density $0$ such that $A + B$ contains all except finitely many integers.
| true |
Er56-4
|
Is there a set $A\subseteq \mathbb{N}$ such that
$$
\lvert A\cap\{1,\ldots,N\}\rvert = o\bigl((\log N)^2\bigr)
$$
and such that every large integer can be written as $p + a$ for some prime $p$ and $a\in A$? Can the bound $O(\log N)$ be achieved? Must such an $A$ satisfy
$$
\liminf_{N\to\infty} \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N} > 1\;?
$$
| false |
Er56-5
|
Let $A\subseteq\mathbb{N}$ be such that every large integer can be written as $n^2 + a$ for some $a\in A$ and $n\ge0$. What is the smallest possible value of
$$
\limsup_{N\to\infty} \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\;?
$$
| false |
Er56-6
|
Let $B\subseteq\mathbb{N}$ be an additive basis of order $k$ with $0\in B$. Is it true that for every $A\subseteq\mathbb{N}$ we have
$$
d_s(A+B)\;\ge\;\alpha+\frac{\alpha(1-\alpha)}{k},
$$
where
$$
\alpha = d_s(A),
\quad
d_s(A) = \inf_{N\ge1}\frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N}
$$
is the Schnirelmann density?
| true |
Er56-7
|
Find the optimal constant $c>0$ such that the following holds. For all sufficiently large $N$, if $A\sqcup B = \{1,\ldots,2N\}$ is a partition into two equal parts with $\lvert A\rvert=\lvert B\rvert=N$, then there is some $x$ such that the number of solutions to
$$
a - b = x
\quad (a\in A,\;b\in B)
$$
is at least $cN$.
| false |
Er56-8
|
We say that $A\subseteq\mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B\subseteq\mathbb{N}$ with $0<d_s(B)<1$. Can a lacunary set $A\subseteq\mathbb{N}$ be an essential component?
| true |
Er56-9
|
Does there exist $B\subseteq\mathbb{N}$ which is not an additive basis, but is such that for every set $A\subseteq\mathbb{N}$ of Schnirelmann density $\alpha$ and every $N$ there exists $b\in B$ such that
$$
\lvert(A\cup(A+b))\cap\{1,\ldots,N\}\rvert\;\ge\;(\alpha+f(\alpha))\,N,
$$
where $f(\alpha)>0$ for $0<\alpha<1$?
| false |
Er56-10
|
Is there an infinite Sidon set $A\subseteq\mathbb{N}$ such that
$$
\lvert A\cap\{1,\ldots,N\}\rvert \;\gg_{\epsilon}\; N^{1/2-\epsilon}
\quad\text{for all }\epsilon>0\;?
$$
| false |
Er56-11
|
Is there $A\subseteq\mathbb{N}$ such that
$$
\lim_{n\to\infty}\frac{(1_A*1_A)(n)}{\log n}
$$
exists and is $\neq0$?
| false |
Er74b-1
|
If $A\subseteq \mathbb{N}$ has $\displaystyle\sum_{n\in A}\frac{1}{n}=\infty$, must $A$ contain arbitrarily long arithmetic progressions?
| false |
Er74b-2
|
We call $m$ practical if every integer $n<m$ is the sum of distinct divisors of $m$. If $m$ is practical, let $h(m)$ be the minimal number of divisors always needed. Are there infinitely many practical $m$ such that
$$
h(m) < (\log\log m)^{O(1)}?
$$
Is it true that
$$
h(n!)<n^{o(1)}\quad\text{or even}\quad h(n!)< (\log n)^{O(1)}?
$$
| false |
Er74b-3
|
Are there infinitely many integers $n,m$ such that
$$
\varphi(n)=\sigma(m)?
$$
| true |
Er74b-4
|
Let $A\subseteq\mathbb{R}$ be infinite. Must there be a set $E\subset\mathbb{R}$ of positive measure which does not contain any dilate–translate $aA+b$ (with $a\neq0$)?
| false |
Er74b-5
|
Let $W(k)$ be the van der Waerden number: for $N\ge W(k)$, any 2-coloring of $\{1,\dots,N\}$ contains a monochromatic $k$-term arithmetic progression. Improve bounds for $W(k)$; for example, prove
$$
W(k)^{1/k}\to\infty.
$$
| false |
Er74b-6
|
Let $N(k,\ell)$ be the minimal $N$ such that any $f:\{1,\dots,N\}\to\{-1,1\}$ has a $k$-term AP $P$ with
$$
\Bigl|\sum_{n\in P}f(n)\Bigr|\ge\ell.
$$
Find good upper bounds for $N(k,\ell)$. Is it true that for all $c>0$ there is $C>1$ with
$$
N(k,ck)\le C^k?
$$
What about
$$
N(k,2)\le C^k\quad\text{or}\quad N(k,\sqrt{k})\le C^k?
$$
| false |
Er74b-7
|
Are there infinitely many $n$ such that, for all $k\ge1$,
$$
\omega(n+k)\ll k\;?
$$
Here $\omega(n)$ is the number of distinct prime divisors of $n$.
| false |
Er74b-8
|
Let $(a_n)$ satisfy $a_n/n\to\infty$. Is the sum
$$
\sum_n\frac{a_n}{2^{a_n}}
$$
irrational?
| false |
Er74b-9
|
Are there infinitely many $n$ for which there exist $t\ge2$ and $a_1,\dots,a_t\ge1$ such that
$$
\frac n{2^n} \;=\;\sum_{k=1}^t \frac{a_k}{2^{a_k}}?
$$
Is this true for all $n$? Does there exist a rational $x$ such that
$$
x=\sum_{k=1}^\infty\frac{a_k}{2^{a_k}}
$$
has $2^{\aleph_0}$ solutions?
| false |
Er74b-10
|
Let $V(x)$ be the count of $n\le x$ for which $\varphi(m)=n$ is solvable. Does
$$
\frac{V(2x)}{V(x)}\to2?
$$
Is there an asymptotic formula for $V(x)$?
| false |
Er74b-11
|
Are there infinitely many positive integers not of the form
$$
n - \varphi(n)\;?
$$
| true |
Er74b-12
|
Is it true that for all sufficiently large $n$ there is $i<n$ such that
$$
p_n^2 < p_{n+i}\,p_{n-i},
$$
where $p_k$ is the $k$th prime?
| true |
Er74b-13
|
Let $A\subset\mathbb{N}$ be the set of $n$ such that for every prime $p\mid n$ there is $d\mid n$ with
$$
d\equiv1\pmod p.
$$
Is it true that for some $c>0$,
$$
\frac{\lvert A\cap[1,N]\rvert}{N}
=\exp\bigl(-(c+o(1))\sqrt{\log N}\,\log\log N\bigr)?
$$
| false |
Er74b-14
|
Let $c(n)$ be minimal so that if $k\ge c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-cubes. Give good bounds for $c(n)$; in particular, is
$$
c(n)\gg n^n?
$$
| false |
Er74b-15
|
Let $h(n)$ be minimal such that $2^n - 1,\,3^n - 1,\,\dots,\,h(n)^n - 1$ are mutually coprime. Does, for every prime $p$, the density $\delta_p$ of integers with $h(n)=p$ exist? Does $\liminf h(n)=\infty$? Is it true that if $p$ is the greatest prime such that $p-1\mid n$ and $p>n^\epsilon$, then $h(n)=p$?
| false |
Er74b-16
|
Let $H(n)$ be the smallest integer $l$ such that there exist $k<l$ with
$$
\gcd(k^n-1,\;l^n-1)=1.
$$
Is it true that $H(n)=3$ infinitely often? Estimate $H(n)$. Is there a constant $c>0$ such that, for all $\epsilon>0$,
$$
H(n)>\exp\bigl(n^{(c-\epsilon)/\log\log n}\bigr)
\quad\text{infinitely often,}
$$
and
$$
H(n)<\exp\bigl(n^{(c+\epsilon)/\log\log n}\bigr)
\quad\text{for all large }n?
$$
Does a similar upper bound hold for the smallest $k$ with $\gcd(k^n-1,2^n-1)=1$?
| false |
Er74b-17
|
Let $g(n)$ count the number of $m$ such that $\varphi(m)=n$. Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ with
$$
g(n)>n^{1-\epsilon}\;?
$$
| false |
Er74b-18
|
Does the set of integers of the form $n+\varphi(n)$ have positive density?
| false |
Er74b-19
|
Let $\alpha\ge1$. Is there a sequence of integers $n_k,m_k$ such that $\tfrac{n_k}{m_k}\to\alpha$ and $\sigma(n_k)=\sigma(m_k)$ for all $k\ge1$, where $\sigma$ is the sum‐of‐divisors function?
| true |
Er74b-20
|
Let $h(x)$ count the number of integers $1\le a<b<x$ with $\gcd(a,b)=1$ and $\sigma(a)=\sigma(b)$. Is it true that
$$
h(x)>x^{2-o(1)}\;?
$$
| false |
Er74b-21
|
Is there an absolute constant $C>0$ such that every integer $n$ with $\sigma(n)>Cn$ is the distinct sum of proper divisors of $n$?
| false |
Er74b-22
|
Are there infinitely many $n$ such that, for all $k\ge1$,
$$
\tau(n+k)\ll k\;?
$$
| false |
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