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Nearest integer to $n/\sqrt{2}$, defined by $\lfloor n/\sqrt{2} + 1/2 \rfloor$. Let $s(n) = \zeta(3) - \sum_{k=1}^n \frac{1}{k^3}$. Conjecture: for $n \ge 1$, $s(a(n)) < \frac{1}{n^2} < s(a(n)-1)$, and the difference sequence of A049473 consists solely of $0$'s and $1$'s, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimber
The ratio $\zeta(4n)/\zeta(2n)^2$ for $n \ge 1$ is the rational number $$ Q_n = -2 \frac{B_{4n}}{B_{2n}^2 \binom{4n}{2n}} $$ where $B_k$ is the $k$-th Bernoulli number. The sequence $a(n)$ is the numerator of $Q_n$, with $a(0)$ defined as $2$. Conjecture: if an integer $n > 1$ is odd, then $\zeta(2n)/\zeta(n)^2$ is irrational. Cf. W. Kohnen (link) and my conjecture in A348829.
$\zeta(5)$ is irrational. 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 provenance Upstream r
This file adds three conjectures about the Mandelbrot and Multibrot sets: - the *MLC conjecture*, stating that these sets are locally connected - the *density of hyperbolicity* conjecture, stating that parameters with attracting cycles are dense in the Mandelbrot and Multibrot sets - the conjecture that the boundaries of these sets have zero area. The first two conjectures are
There are infinitely many prime numbers of the form `k * 2 ^ k - 1` for `k > 1`. 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 be
**Andrica's conjecture** The inequality $\sqrt{p_{n+1}}-\sqrt{p_n} < 1$ holds for all $n$, where $p_n$ is the $n$-th prime 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 the pinned Lean 4.33.1 environmen
The second Cuboid 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 and must be accepted by the deployed verifier before it is available. Sources and provenance Upstrea
**The Tu-Deng conjecture.** For $k \ge 2$ and a nonzero residue $t$ modulo $2^k - 1$, there are at most $2^{k-1}$ pairs of residues $(a, b)$ with $a + b = t$ whose binary weights (of their representatives in $0, \dots, 2^k - 2$) sum to at most $k - 1$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f7
The value $a(n)$ is given by $$ a(n) = \sum_{p \mid n, p \text{ prime}} 2^{\pi(p) - 1} $$ where $\pi(p) = \mathrm{primeCounting}(p)$ gives the 1-based index of the prime $p$. Starting at any $n$ and iterating the map $n \mapsto a(n)$, we will always reach $0$. - _Antti Karttunen_, Jun 18,20 2017 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectu
The game **Catch-Up** (Isaksen–Ismail–Brams–Nealen, 2015) is a two-player, perfect-information game played on a finite nonempty set `S` of positive integers. Each time a player removes a number from `S`, that number is added to the player’s score. **Rules.** * The scores start at `0`. Player `p1` starts by removing **exactly one** number from `S`. * After the first move, player
The sequence $a(n)$ is the smallest positive integer $m$ such that $n \cdot 2^m + 1$ is prime, or $0$ if no such $m$ exists. There is a conjecture that the first zero is $n = 65536 = 2^{16}$ (which is equivalent to the statement that $2^{2^k} + 1$ is composite for $k > 4$). - _T. D. Noe_, Feb 25 2011 Mathematical status Open: marked `research open` in google-deepmind/formal-con
Catalan-Mersenne numbers: $a(0) = 2$; for $n \ge 0$, $a(n+1) = 2^{a(n)} - 1$. Catalan-Mersenne conjecture: All terms of the Catalan-Mersenne sequence are prime. 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
For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the equation $ax^n - by^m = c$ where $(m, n) \neq (2, 2)$ and $x, y > 1$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop definition is supp
There exists a positive number $C$ such that for any integer $x, y$ with $y^2 \ne x^3$, $|y^2 - x^3| > C \sqrt{|x|}$. Original Hall's conjecture with exponent $1/2$. 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 fo
The number of distinct products (including the empty product 1) of any subset of $\{1, 2, \dots, n\}$. Conjecture: let $p \le n$ be prime. If $m$ and $p^a m$ are two such products, then so is $p^k m$ for all $0 < k < a$. - Yan Sheng Ang, Feb 13 2020 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba
Let G be a finite bridgeless cubic graph (every vertex has degree 3 and no edge is a bridge). Do there exist six perfect matchings M₁, …, M₆ of G, repetitions allowed, such that every edge of G belongs to exactly two of them? Why it matters Open Problem Garden rates the problem "Outstanding" and notes it holds trivially for 3-edge-colourable cubic graphs, so the conjecture asse
For every positive integer k, does there exist a square matrix H of order n = 4k, with every entry either +1 or −1, whose distinct rows have dot product zero? Equivalently, H Hᵀ = nI. The pinned Lean target uses the equivalent maximal-determinant condition |det H| = n^(n/2) and also includes the harmless empty matrix at k = 0, with determinant 1. Why it matters Orthogonal sign
A positive integer n is perfect if the sum of its positive divisors smaller than n equals n. For example, 6 = 1 + 2 + 3. Is every perfect number even? Equivalently, does no odd perfect number exist? This target does not ask whether there are infinitely many perfect numbers. Why it matters Even perfect numbers have the Euclid–Euler description in terms of Mersenne primes. The od
For integers k ≥ 6 and n ≥ 2, let a₁, …, aₙ be positive integers and b a nonnegative integer. If a₁^k + … + aₙ^k = b^k, must n ≥ k? The summands need not be distinct. Positivity makes b = 0 impossible. This is only the k ≥ 6 restriction of Euler’s original conjecture, not the disproved unrestricted statement. Why it matters Counterexamples to the original conjecture show why an
For a positive integer n, define C(n) = n/2 when n is even and C(n) = 3n + 1 when n is odd. Starting from n, repeatedly apply C. Is it true that for every n ≥ 1 there is an integer m ≥ 0 for which C iterated m times sends n to 1? The starting value counts as the zeroth iterate; negative integers and zero are not part of the question. Why it matters One elementary rule produces