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Any integer $n \geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers and $z \leq w$, such that both $x$ and $x + 24y$ are squares. Zhi-Wei Sun has offered a \$2,400 prize for the first proof. **Zhi-Wei Sun's Conjecture (A281976)**: Any integer $n \geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers and $z \l
Any nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square. Zhi-Wei Sun has offered a prize of 1,680 RMB for the first proof. **Zhi-Wei Sun's 1680-Conjecture (A280831)**: Any nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 +
Any integer $n > 1$ can be written as $a^2 + b^2 + 3^c + 5^d$ where $a, b, c, d$ are nonnegative integers. Zhi-Wei Sun has offered a \$3,500 prize for the first proof. **Zhi-Wei Sun's Conjecture (A303656)**: Any integer $n > 1$ can be written as the sum of two squares, a power of 3, and a power of 5. Mathematical status Open: marked `research open` in google-deepmind/formal-con
Conjecture: for $n > 3$, $\textrm{numerator}(-2/n + \sum_{k=1}^{n} \frac{2^k}{k}) == 0 (\textrm{mod} n^2)$ if and only if n is prime. Conjecture: for $n > 3$, $\textrm{numerator}(-2/n + \sum_{k=1}^{n} \frac{2^k}{k}) == 0 (\textrm{mod} n^2)$ if and only if n is prime. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db463473
Conjecture: Except for the first term all terms are even. 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 accepted by the deployed verifier before it is available. S
Alternatively the sequence `a` can be defined as $a(n) = \sum_{k=0}^n \binom{n+k-1}{k} F(n-k+1)$, where $F(m)$ is the $m$-th Fibonacci number. We formalize a conjecture about the number of words of length $n$ in a set $X$ being related to this sequence. "The number of words of length $n$ for $n \le 12$ is given by $a(n+1)$. Is this always true?" Formalized as $|X_n| = a(n-1)$ f
Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime. Zhi-Wei Sun has offered a \$200 prize for the first proof. **Zhi-Wei Sun's Conjecture (A232174)**: Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime. Mathematical status Open: marked `research open` in g
Number of ways to write $n = x+y$, for $x,y > 0$ such that $2^x + y$ is prime. Zhi-Wei Sun has offered a \$1000 prize for the first proof. The conjecture for sequence A231201: for any $n > 1$, there exist $x, y > 0$ such that $n = x + y$ and $2^x + y$ is prime. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b
$a(n)$ is the least $k$ such that all of $k, 2k+1, 3k+2, ..., nk+n-1$ are primes, or $0$ if no such $k$ is found. It is conjectured $k$ always exists. It is conjectured k always exists. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop defin
Every prime $p$ has a primitive root $0 < g < p$ of the form $k^2 + 1$, where $k$ is an integer. Zhi-Wei Sun has offered a prize of RMB 2,000 for the first proof. **Zhi-Wei Sun's Conjecture (A239957)**: Every prime $p$ has a primitive root $0 < g < p$ of the form $k^2 + 1$, where $k$ is an integer. Mathematical status Open: marked `research open` in google-deepmind/formal-conje
The sequence is given by the combinatorial identity: $a(n) = \sum_{k = 0}^n \binom{n}{k} \binom{2n+k-1}{n-1}$ with $a(0) = 1$. The recurrence given below can be rewritten in the form $$(2n+1)(2n+2)P(2,n)a(n+1) - (2n-1)(2n-2)P(2,-n)a(n-1) = Q(2,n^2)a(n),$$ where the polynomial $Q(2,n) = 4(55n^2 - 34n + 3)$ and the polynomial $P(2,n) = 5n^2 - 5n + 1$ satisfies the symmetry condit
For statistical reasons it is conjectured that the sequence is finite. Also it is conjectured that $a(40)$ does not exist (i.e. the sequence is empty for $n=40$). For statistical reasons it is conjectured that the sequence is finite. This is formalized as the assertion that for large enough $n$, no valid zeroless power exists, which in our definition results in $a(n) = 0$. Math
Conjecture: all items for $n \ge 4$ are greater than or equal to $1$. This is a stronger conjecture than the Goldbach conjecture. 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
The sequence is defined by $a(0) = 0$, $a(1) = 1$, and for $n \ge 0$, $$a(n+2) = (a(n+1) \mathbin{\mathrm{XOR}} (n+2)) - a(n),$$ where $\mathrm{XOR}$ is the bitwise exclusive-or operator on integers. Conjecture: the sequence contains 8 zeros. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dd
The sequence is defined by $$a(n) = \mathrm{prime}(n) \cdot \mathrm{prime}(n+1) \bmod \mathrm{prime}(n+2),$$ where $\mathrm{prime}(k)$ is the $k$-th prime number ($\mathrm{prime}(1)=2$). Conjecture: For $x > 10^9$, the most frequent value in $a(n)$, $n=1\dots x$, has form $120k$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision
Erdős was unable to prove that if the two products have the same factors then there must exist a prime between $n_1$ and $n_2$. 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 a
The sequence $a(n)$ has exponential generating function $$E(x) = \prod_{k=1}^\infty \left(1 - \frac{x^k}{k!}\right),$$ so that $a(n) = n! [x^n] \prod_{k=1}^n \left(1 - \frac{x^k}{k!}\right)$. $a(n)$ differs in sign from $a(n-1)$ if and only if $n$ is a triangular number (checked up to $n = 1225 = (50 \cdot 51)/2$). - _Peter Bala_, Mar 17 2022 Mathematical status Open: marked `r
Milićević [ElJo23, Conjecture 11.1] conjectures the following 2-adic variant. For any $k \in \mathbb{N}$, there exists $K = K(k)$ such that the following is true. Let $r$ be a positive integer, and let $a_1, \dots, a_k \in \mathbb{Z}/2^r\mathbb{Z}$. Let $d$ be the largest integer such that $\sum_{i \in I} a_i \equiv 0 \pmod{2^d}$ for some non-empty subset $I \subset [k]$. Then
Conjecture p.579 in [Aa19]: $\left(\frac{1}{3} + o(1)\right) n^2$. 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 accepted by the deployed verifier before it is ava
Let G be a finite bridgeless cubic graph. Do there exist three perfect matchings M₁, M₂, M₃ of G with empty common intersection, i.e. no edge of G belongs to all three? Why it matters Fouquet and Vanherpe (arXiv:0809.4821) record the conjecture as due to Fan and Raspaud and prove a minimum counterexample must have at least 32 vertices; Open Problem Garden notes it would follow