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The conjecture says that every normally generating `n`-tuple in the free group on `n` generators is Andrews-Curtis equivalent to the standard free basis. **The Andrews-Curtis conjecture.** Every normally generating `n`-tuple in the free group of rank `n` is Andrews-Curtis equivalent to the standard tuple of free generators. Mathematical status Open: marked `research open` in go
Any integer $n > 0$ can be written as $\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ with $w, x, y, z$ nonnegative integers. Zhi-Wei Sun has offered a $2,468 prize for the first proof (or $2,468 RMB for a counterexample). The conjecture has been verified for all $n$ up to $1.2 \times 10^{12}$ by Yaakov Baruch (March 2019). **Zhi-Wei Sun's 2-4-6-8 Conjecture
Least positive multiple of $n$ that when written in base 10 uses only 0's and 1's. It is known that $a(10^k - 1) = (10^{9k} - 1) / 9$ for all $k$. Is $a(n) < a(10^k - 1)$ for all $n < 10^k - 1$? - David Radcliffe, Aug 01 2025 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-
Conjecture: $\liminf_{n \to \infty} \frac{a(n)}{p_{n+1}^2} = 1 <$ $\limsup_{n \to \infty} \frac{a(n)}{p_{n+1}^2} = 2$. - Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is
The Bézier-type Bernstein operators $B_{n,\alpha}$ for $\alpha > 0$ are defined for $f : [0,1] \to \mathbb{R}$ by $$ (B_{n,\alpha} f)(x) = \sum_{k=0}^n f\!\left(\frac{k}{n}\right) \left( J_{n,k}(x)^{\alpha} - J_{n,k+1}(x)^{\alpha} \right), $$ where $$ J_{n,k}(x) = \sum_{j=k}^n p_{n,j}(x), \qquad p_{n,j}(x) = \binom{n}{j} x^j(1-x)^{n-j}, $$ and $J_{n,n+1}(x) = 0$. In the classic
A binomial coefficient sum: $$a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \left( \binom{n}{k} - \binom{n}{k-1} \right)^3$$ where $\binom{n}{-1} = 0$. Let $b(n) = a(2n-1)$. Then the supercongruence $b(n p^k) \equiv b(n p^{k-1}) \pmod{p^{3k}}$ holds for positive integers $n$ and $k$ and all primes $p \ge 5$. - Zhi-Wei Sun, Nov 16 2019 Mathematical status Open: marked `research open`
For any `k ≥ 2`, let `a₁,...,aₖ` and `b₁,...,bₖ` be integers with `aᵢ > 0`. Suppose that for every prime `p` there exists an integer `n` such that `p ∤ ∏ i, (aᵢ n + bᵢ)`. Then there exist infinitely many `n` such that `aᵢ n + bᵢ` is prime for all `i`. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77
For any tree $T$ with $n$ edges, the complete graph $K_{2n+1}$ decomposes into $2n+1$ edge-disjoint copies of $T$. A "copy" of $T$ is the image $T.\text{map}(f_i)$ of $T$ under a vertex embedding $f_i : V \hookrightarrow \text{Fin}(2n+1)$; the copies are pairwise edge-disjoint and together cover every edge of $K_{2n+1}$. Mathematical status Open: marked `research open` in googl
OEIS A002407 lists the primes that are differences of two consecutive positive cubes. The sequence is conjectured to be infinite. This sequence is believed to be infinite. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is suppl
A binomial coefficient summation: $a(n) = S(3, n) / S(1, n)$, where for a positive integer $r$ we define $$S(r,n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \left( \binom{n}{k} - \binom{n}{k-1} \right)^r$$ with $\binom{n}{-1} = 0$. Let $b(n) = a(2n-1)$. Then the supercongruence $b(n p^k) \equiv b(n p^{k-1}) \pmod{p^{3k}}$ holds for positive integers $n$ and $k$ and all primes $p \ge 5$
Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ where $a, b, c, d, x, y$ are nonnegative integers. Zhi-Wei Sun has offered a \$2,500 prize for the first proof. **Zhi-Wei Sun's Four-Square Conjecture (A308734)**: Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d
The determinant of the $n \times n$ Hankel matrix whose entries are the first $2n-1$ prime numbers. The matrix $M$ has entries $M_{i, j} = p_{i+j}$ for $i, j \in \{0, \dots, n-1\}$, where $p_k = \mathrm{Nat.nth\;Nat.Prime} (k)$ is the $k$-th prime starting at $p_0=2$. $a(0)=1$ by convention. "I conjecture that $a(4)$ is the only zero. - _Jon Perry_, Mar 22 2004" Mathematical st
The Beaver Math Olympiad (BMO) is a set of mathematical reformulations of the halting/nonhalting problem of specific Turing machines from all-0 tape. These problems came from studying small Busy Beaver values. Some problems are open and have a conjectured answer, some are open and don't have a conjectured answer, and, some are solved. Among these problems is the Collatz-like *A
The negation of `Finite.Equation677_implies_Equation255`. Probably this is true. It would be a stronger form of `Equation677_not_implies_Equation255`. Discussion thread here: https://leanprover.zulipchat.com/#narrow/channel/458659-Equational/topic/FINITE.3A.20677.20-.3E.20255 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d
**Markel's $S_3$-conjecture** (1973): any nontrivial finite ah-group is isomorphic to $S_3$. The conjecture is open in general; it is known to be true for solvable groups. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is suppl
Central factorial numbers: $a(n) = 4^n (n!)^2 = ((2n)!!)^2$. Let $\zeta$ be a primitive $(2n+1)$-th root of unity. Then the permanent of the $2n \times 2n$ matrix $[m(j,k)]_{j,k=1..2n}$ is $a(n)/(2n+1) = ((2n)!!)^2/(2n+1)$, where $m(j,k)$ is $1$ or $(1+\zeta^{j-k})/(1-\zeta^{j-k})$ according as $j = k$ or not. - Zhi-Wei Sun, Dec 21 2021 Mathematical status Open: marked `researc
Integral factorial ratio sequence: $$a(n) = \frac{(30n)! n!}{(15n)! (10n)! (6n)!}$$ Supercongruence: "a(p^k) == a(p^(k-1)) ( mod p^(3*k) ) for any prime p >= 5 and any positive integer k." - _Peter Bala_, Jan 24 2020 More generally, "the congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) may hold for any prime p >= 5 and any positive integers n and k." Mathematical status Ope
Central trinomial coefficients: largest coefficient of $(1 + x + x^2)^n$, which is the coefficient of $x^n$ in the expansion of $(1 + x + x^2)^n$. An integer $n > 3$ is prime if and only if $a(n) \equiv 1 \pmod{n^2}$. We have verified this for $n$ up to $8 \cdot 10^5$, and proved that $a(p) \equiv 1 \pmod{p^2}$ for any prime $p > 3$ (cf. A277640). - Zhi-Wei Sun, Nov 30 2016 Mat
Smallest prime $p$ such that $p + n$ is an $n$-th power, or $0$ if no such number exists. That is, the smallest prime of the form $k^n - n$. Conjecture: if a(k) = 0 then k is an even square. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop
The number of squares modulo $n$. This is the cardinality of the set $\{k^2 \bmod n \mid k \in \{0, 1, \dots, n-1\}\}$. $n^2 \equiv 1 \pmod{a(n)(a(n)-1)}$ if and only if $n$ is an odd prime. - Thomas Ordowski, Jun 08 2017 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-1