Browse Problems

Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable. Erdős proved V(x)=x(logx)^(−1+o(1)). Ref: Erdős, P., _On the normal number of prime factors of $p-1$ and some related problems concerning …

Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable. `V(x) ≍ x/log x*e^(C_1*(log log log x − log log log log x)^2+C_2 log log log x − C_3 log …

Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable. `V(x)=x/logx * e^((C+o(1))(log log log x)^2)`, for some explicit constant `C>0`. Ref:Maier, Helmut and Pomerance, Carl, _On the number of …

Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable. Pillai proved `V(x)=o(x)`. Ref: S. Sivasankaranarayana Pillai, _On some functions connected with $\phi(n)$_

Let\[V'(x)=\#\{\phi(m) : 1\leq m\leq x\}\]and\[V(x)=\#\{\phi(m) \leq x : 1\leq m\}.\] Does $\lim V(x)/V'(x)$ exist? Formalization note: We formalize the limit of the inverse fraction V'(x)/V(x) to ensure the limit is finite (bounded …

Is it $>1$?

Are there infinitely many integers not of the form $n - \phi(n)$? This is true, as shown by Browkin and Schinzel [BrSc95]. [BrSc95] Browkin, J. and Schinzel, A., _On integers not of …

It follows from a slight strengthening of the Goldbach conjecture that every odd number can be written as $n - \phi(n)$. In particular, we assume that every even number greater than 6 …

It seems to be open whether there is a positive density set of integers not of the form $n - \phi(n)$.

Erdős has shown that a positive density set of integers cannot be written as $\sigma(n) - n$. [Er73b] Erdős, P., _Über die Zahlen der Form $\sigma (n)-n$ und $n-\phi(n)$_. Elem. Math. (1973), …

A solution to erdos_418 was shown by Browkin and Schinzel [BrSc95] by showing that any integer of the form $2^(k + 1)\cdot 509203$ is not of the form $n - \phi(n)$. [BrSc95] …

**Erdős Problem 42**: Let M ≥ 1 and N be sufficiently large in terms of M. Is it true that for every maximal Sidon set `A ⊆ {1,…,N}` there is another Sidon …

Is there a sequence $1 \le d_1 < d_2 < \dots$ with density 1 such that all products $\prod_{u \le i \le v} d_i$ are distinct?

Does $f(n)$ miss infinitely many integers?

Does $f$ become stationary at some point?

How does $f$ grow?

Is $f$ surjective?

Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \ldots, a_n$ all possible values of $a_i a_j - 1$ with $i \neq j$. Is it …

**Erdős Problem 427**: is it true that, for every $n$ and $d$, there exists $k$ such that $$ d \mid p_{n + 1} + \cdots + p_{n + k}, $$ where $p_r$ …

Cedric Pilatte has observed that a positive solution to Erdős Problem 427 follows from Shiu's theorem.

**Shiu's theorem**: for any $k \geq 1$ and $(a, q) = 1$ there exist infinitely many $k$-tuples of consecutive primes $p_m, \dots, p_{m + k - 1}$ all of which are congruent …

Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and \[\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?\]

A variant asking for explicit bounds on how large N needs to be in terms of M. This version provides a constructive function f such that for all M ≥ 1 and …

Let $k \le n$. What choice of $A\subseteq\{1, \dots, n\}$ of size $|A| = k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with …

Let $k \le n$. Out of all $A\subseteq\{1, \dots, n\}$ of size $|A| = k$, does $A = \{n, n - 1, \dots, n - k + 1\}$ maximise the number of …

**Erdős Problem 44:** Let N ≥ 1 and `A ⊆ {1,…,N}` be a Sidon set. Is it true that, for any ε > 0, there exist M = M(ε) and `B ⊆ …

Let $\operatorname{Log} x := \max\{\log x, 1\}$, $\operatorname{Log}_2x = \operatorname{Log} (\operatorname{Log} x)$, and $\operatorname{Log}_3x = \operatorname{Log}(\operatorname{Log}(\operatorname{Log} x)).$ Is it true that if $A\subseteq\mathbb{N}$ is such that $$ \frac{1}{\operatorname{Log}_2 x} \sum_{n\in A: n\leq …

Tao resolved erdos_442 in the negative in Theorem 1 of https://arxiv.org/pdf/2407.04226. The following is a formalisation of that theorem with $C_0 = 1$. Let $\operatorname{Log} x := \max\{\log x, 1\}$, $\operatorname{Log}_2x = …

The case where we start with an empty set (constructing large Sidon sets).

Is it true that `limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤`?

`limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop ≥ 2`, and this is proved in [Po79].

Let `q : ℕ → ℕ` be a strictly increasing sequence of primes such that `q (n + 2) - q (n + 1) ≥ q (n + 1) - q n`. …

Let `q : ℕ → ℕ` be a strictly increasing sequence of primes such that `q (n + 2) - q (n + 1) ≥ q (n + 1) - q n`. …

Is there some $\epsilon > 0$ such that there are infinitely many $n$ where all primes $p \le (2 + \epsilon) \log n$ divide $$ \prod_{1 \le i \le \log n} (n …

Taking $n$ to be the product of primes between $\log n$ and $(2 + o(1)) \log n$ gives an example where $$ q(n, \log n) \ge (2 + o(1)) \log n. $$ …

More generally, let $q(n, k)$ denote the least prime which does not divide $\prod_{1 \le i \le k}(n + i)$. This problem asks whether $q(n, \log n) \ge (2 + \epsilon) \log …

Let $\operatorname{lcm}(1, \dots, n)$ denote the least common multiple of $\{1, \dots, n\}$. Let $p_k$ be the $k$-th prime. Is it true that for all $k \geq 1$, $\operatorname{lcm}(1, \dots, p_{k+1}-1) < …

Is there a function $f$ with $f(n)\to\infty$ as $n\to\infty$ such that, for all large $n$, there is a composite number $m$ such that $$ n + f(n) < m < n + …

Let $A$ be the set of all $n$ such that $n = d_1 + ⋯ + d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m …

Are there any odd weird numbers?

Are there infinitely many primitive weird numbers?

If there are no odd weird numbers then every weird number has abundancy index < 4.

Fang [Fa22](https://arxiv.org/abs/2207.12906) has shown there are no odd weird numbers below $10^{21}$.

Liddy and Riedl [LiRi18](https://ideaexchange.uakron.edu/honors_research_projects/728/) have shown that an odd weird number must have at least 6 prime divisors.

Melfi [Me15](https://mathscinet.ams.org/mathscinet/relay-station?mr=3276337) has proved that there are infinitely many primitive weird numbers, conditional on the fact that $p_{n+1} - p_n < \frac{1}{10} \sqrt{p_n}$ for all large $n$, which in turn would follow …

Benkoski and Erdős [BeEr74](https://mathscinet.ams.org/mathscinet/relay-station?mr=347726) proved that the set of weird numbers has positive density.

Is there a polynomial $f:\mathbb{Z}\to \mathbb{Z}$ of degree at least $2$ and a set $A\subset \mathbb{Z}$ such that for any $z\in \mathbb{Z}$ there is exactly one $a\in A$ and $b\in \{ f(n) …

There is no such $A$ for any polynomial $f(x) = aX^2 + bX + c$, if $a | b$ with $a \ne 0$ and $b \ne 0. This was found be AlphaProof …

Probably there is no such $A$ for the polynomial $X^k$ for any $k \ge 2$. This is asked in [Sek59].

There is no such $A$ for the polynomial $f(x) = X^2$. This is shown in [Sek59].