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Starting from an integer $n\geq 2$, list its prime factors in nondecreasing order with multiplicity, concatenate their decimal representations, and repeat. The home-prime conjecture says that this process always reaches a prime. For example, $$25 \longmapsto 55 \longmapsto 511 \longmapsto 773.$$ Every integer at least two reaches a home prime. Mathematical status Open: marked `
Any nonnegative integer can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ with $x, y, z$ nonnegative integers. Zhi-Wei Sun has offered a USD 135 prize for the first proof of this conjecture. **Zhi-Wei Sun's Conjecture (A287616)**: Any nonnegative integer can be written as the sum of a triangular number $x(x+1)/2$, a generalized pentagonal number $y(3y+1)/2$, and a generalize
The multiplicative order of 2 modulo $2n+1$. In other words, the least $m > 0$ such that $2n+1$ divides $2^m - 1$. If $p$ is an odd prime then $a((p^3-1)/2) = p \cdot a((p^2-1)/2)$. Because otherwise $a((p^3-1)/2) < p \cdot a((p^2-1)/2)$ iff $a((p^3-1)/2) = a((p-1)/2)$ for a prime $p$. Equivalently $p^3$ divides $2^{p-1}-1$, but no such prime $p$ is known. - Thomas Ordowski, Fe
$a(n)$ is the number of odd primes $p$ between $n^2$ and $(n+1)^2$ such that the Legendre symbol $\left(\frac{n}{p}\right) = 1$. Conjecture: $a(n) > 0$ for all $n > 0$. - _Zhi-Wei Sun_, Dec 29 2012 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability
The sequence is defined by $a(1) = 2$, and for $n \ge 2$, $$(2n+1)^3 a(n) = 32n^3 a(n-1) + (21n^3 + 22n^2 + 8n + 1) \binom{2n-1}{n}^4.$$ Each term $a(n)$ is a positive integer. - _Zhi-Wei Sun_, Apr 06 2010 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal avail
An Abelian cube is a string of the form $x x' x''$ with $|x| = |x'| = |x''|$ and $x$ is a permutation of $x'$ and $x''$. The number of Abelian cubes of length $3n$ over an alphabet of size 3 is given by $$a(n) = \sum_{k=0}^n \binom{n}{k}^3 \sum_{j=0}^k \binom{k}{j}^3.$$ Conjecture: the supercongruences $a(n \cdot p^k) \equiv a(n \cdot p^{k-1}) \pmod{p^{3k}}$ hold for primes $p
$a(n) = \min \{k \in \mathbb{N} \mid 0 < k \wedge \text{Prime}(|\Phi_k(n)|) \}$, where $\Phi_k(n)$ is the $k$-th cyclotomic polynomial evaluated at $n$. Is $a(n)$ defined for all $n \ge 1$? That is, for every $n \ge 1$, does there exist $k > 0$ such that $|\Phi_k(n)|$ is prime? Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd
A Steiner system $S(t, k, n)$ is a collection of $k$-element subsets (called blocks) of an $n$-element set such that every $t$-element subset is contained in exactly one block. Construct an $S(t, k, n)$-Steiner system with $n > k > t > 5$, $t < 10$, and $n < 200$. No example of a Steiner system with $t > 5$ is known, despite a 2014 existence theorem by Keevash showing that such
The Pierce-Birkhoff conjecture states that for every real piecewise-polynomial function `f : ℝⁿ → ℝ`, there exists a finite set of polynomials `gᵢⱼ ∈ ℝ[x₁, ..., xₙ]` such that `f = supᵢ infⱼ(gᵢⱼ)`. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability
Given any set of $n$ complex numbers $\{z_1, ..., z_n\}$ that are linearly independent over $\mathbb{Q}$, the field extension $\mathbb{Q}(z_1, ..., z_n, e^{z_1}, ..., e^{z_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (check
An integer `n : ℤ` is `(m,k)-perfect` if `σᵐ(n) = kn` where `σᵐ` is the mᵗʰ iterate of the sum of divisors function. There does not exist a $(2,5)$-perfect number 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 t
**PSW conjecture** (Selfridge's test) Let $p$ be an odd number, with $p \equiv \pm 2 \pmod{5}$, $2^{p-1} \equiv 1 \pmod{p}$ and $F_{p+1} \equiv 0 \pmod{p}$, then $p$ is a prime number. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop defini
A Pierpont prime is a prime of the form $2^a 3^b + 1$, where $a$ and $b$ are nonnegative integers. Marc Gleason conjectured that there are infinitely many. There are infinitely many Pierpont primes. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability
The Komlós conjecture in discrepancy theory: there is a universal constant $K$ such that for all $n, m$ and all vectors $v\_1, \dots, v\_n \in \mathbb{R}^m$ with $\|v\_i\|\_2 \le 1$, there exist signs $\varepsilon\_i \in \{-1, +1\}$ such that $$\left\|\sum\_{i=1}^n \varepsilon\_i v\_i\right\|\_\infty \le K.$$ The best known bound is due to Banaszczyk, who proved that one can al
There are infinitely many prime Pell numbers 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. Sources and pr
For every integer $x \ge 2$ there exists a prime between $x(x-1)$ and $x^2$. 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
Kummer–Vandiver conjecture states that for every prime $p$, the class number of the maximal real subfield of $\mathbb{Q}(\zeta_p)$ is not divisible by $p$. - 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 pi
Now form a sequence beginning with any positive integer, where each subsequent term is obtained by applying the operation defined above to the previous term. The **Juggler Conjecture** states that for any positive integer $n$, there exists a natural number $m$ such that the $m$-th term of the sequence is $1$. Mathematical status Open: marked `research open` in google-deepmind/f
The **Fermat–Catalan conjecture** states that the equation $a^m + b^n = c^k$ has only finitely many solutions $(a,b,c,m,n,k)$ with distinct triplets of values $(a^m, b^n, c^k)$ where $a, b, c$ are positive coprime integers and $m, n, k$ are positive integers satisfying $\frac 1 m + \frac 1 n + \frac 1 k < 1$. Mathematical status Open: marked `research open` in google-deepmind/f
The **Inverse Galois Problem**: every finite group is isomorphic to the Galois group of a Galois extension of the rationals. 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