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The sequence $a(n)$ is the number of prime powers $k$ strictly between the $n$-th prime $p_n$ and the $(n+1)$-th prime $p_{n+1}$: $p_n < k < p_{n+1}$. It is conjectured that $a(n) \le 2$ for all $n$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availabilit
The area of a regular $n$-gon with side length 1 is given by $\frac{n}{4} \cot(\pi / n) = \frac{n}{4 \tan(\pi / n)}$. "Usually (perhaps always?) $\lfloor n^2 / (4\pi) - \pi / 12 \rfloor$ for a polygon of circumference $n$. Note that the area of a circle with circumference $C$ is $C^2 / (4\pi)$." Mathematical status Open: marked `research open` in google-deepmind/formal-conjectu
There are infinitely many real quadratic fields `ℚ(√d)` with class number one, where `d > 1` is a squarefree integer. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be
The sequence $a(n)$ satisfies $\sum_{k \mid n} \frac{a(k)}{k!} = \sum_{j=1}^n \frac{1}{j} = H_n$, where the sum on the left is over positive divisors $k$ of $n$. By Möbius inversion, $$a(n) = n! \sum_{d \mid n} \mu(n/d) H_d$$ where $H_d = \sum_{j=1}^d \frac{1}{j}$ is the $d$-th harmonic number. The terms are not all positive. The first negative one is $a(30) = -2269064464730281
A repunit is a number whose digits in some base are all $1$. Here a nontrivial representation has at least three digits. The Goormaghtigh conjecture says that $31$ and $8191$ are the only numbers having nontrivial repunit representations in two different bases. The only Goormaghtigh numbers are $31$ and $8191$. Mathematical status Open: marked `research open` in google-deepmind
The sequence $a(n)$ is the number of times the binary expansion of $n$ appears as a contiguous sublist (infix) in the binary expansion of $n^2$. Is $a(n) \le 1$ for all $n$? Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is sup
The Elliott–Halberstam conjecture: for every $\theta < 1$ and $A > 0$ there exists a constant $C > 0$ such that $$\sum_{1 \le q \le x^{\theta}} E(x; q) \le \frac{C x}{\log^A x}$$ for all $x > 2$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A
The sequence $a(n)$ gives the smallest $k$ such that the decimal expansion of $k!$ contains exactly $n$ occurrences of the digit '6', or $0$ if no such $k$ exists. It is conjectured that $a(24) = 0$ since no factorial less than $10000$ contained just 24 sixes. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2
The $n$-th term $a(n)$ is given by $$a(n) = \sum_{k=0}^{n-1} \frac{1}{k+1} \binom{2k}{k} \binom{k}{n-1-k}$$ Conjecture: for $n > 0$, $a(n)$ is also the number of sequences of length $n - 1$ covering an initial interval of positive integers and avoiding three terms $(\dots, x, \dots, y, \dots, z, \dots)$ such that $x \le y \le z$. - Gus Wiseman, Jun 17 2021 Mathematical status O
For positive integers $k$ and $m$, let $$S_k(m)=1^k+2^k+\cdots+(m-1)^k.$$ The Erdős–Moser conjecture says that $S_k(m)=m^k$ has only the solution $(k,m)=(1,3)$. The only positive solution of $S_k(m)=m^k$ is $(k,m)=(1,3)$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-1
The Catalan-Larcombe-French sequence defined by $a(0)=1$, $a(1)=8$, and $$n^2 a(n) = 8(3n^2 - 3n + 1) a(n-1) - 128(n-1)^2 a(n-2)$$ for $n \ge 2$. Conjecture: let $P(n)$ be the $(n+1) \times (n+1)$ Hankel-type determinant with $(i,j)$-entry equal to $a(i+j)$ for all $i,j = 0, \ldots, n$. Then $P(n)/2^{n(n+3)}$ is a positive odd integer. - Zhi-Wei Sun, Aug 14 2013 Mathematical st
The sequence $a(n)$ counts the number of integers $s \in \{1, \dots, n-1\}$ such that $n^2 + s^2$ is prime: $$a(n) = \sum_{s=1}^{n-1} [\text{Prime}(n^2 + s^2)]$$ Conjecture: $a(n) > 0$ for all $n > 1$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availabil
A067720 lists numbers $k$ such that $\varphi(k^2 + 1) = k \cdot \varphi(k + 1)$, where $\varphi$ is Euler's totient function. The sequence exhibits a strong connection to primes: for almost all terms $k$, $k + 1$ is prime. The conjecture states that $k = 8$ is the only exception. For members of the sequence other than $8$, we have $k + 1$ is prime. Mathematical status Open: mar
**Büchi's problem (first open case, $M = 5$)**: For all integers $x$ and $a$, if $(x+n)^2 + a$ is a perfect square for $n = 0, 1, 2, 3, 4$, then $a = 0$. Non-trivial sequences of length 3 and 4 are known to exist, so $M = 5$ is the first open case. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3
Number of primes strictly less than $n^2$. Conjecture: all the numbers $\sum_{i=j}^k \frac{1}{a(i)}$ with $1 < j \le k$ have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop def
For every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that $φ(n) = φ(m)$ where $φ$ is the Euler totient function. *Carmichael's totient function conjecture*: For every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that $φ(n) = φ(m)$. Mathematical status Open: marked `research open` in google-deepmind/form
The $n$-th prime number $p_n$, where $p_1 = 2$. **Conjecture from Thomas Ordowski (2023)**: $\log \log a(n+1) - \log \log a(n) < 1/n$ for $n > 0$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is supplied for the pinned Lean 4
The smallest positive integer $m$ whose Euler totient equals $n!$. Conjecture: unless $n! + 1$ is prime (i.e., $n \in \text{A002981}$), $a(n) = p q$ where $p$ is the least prime $> \sqrt{n!}$ such that $(p - 1) \mid n!$ and $q = \frac{n!}{p - 1} + 1$ is prime. - M. F. Hasler, Oct 04 2009 We assume $a(n) \ne 0$ and $(p(n)).\text{Prime}$ to ensure the `sInf` searches are non-empt
The difference between the smallest prime strictly greater than $n^2$ and $n^2$. Conjecture: $a(n) \le 1 + \phi(n)$ for $n > 0$. This improves on Oppermann's conjecture, which says $a(n) < n$. - Thomas Ordowski, Dec 17 2014 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09
The **Beal Conjecture**: if we are given positive integers $A, B, C, x, y, z$ such that $x, y, z > 2$ and $A^x + B^y = C^z$ then $A, B, C$ have a common divisor. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is supplied for th