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Let $P = (m_1, \dots, m_k)$ be a tuple of positive even integers. Let $\pi_P(n)$ denote the number of primes $p\leq n$ such that $(p, p + m_1, \dots, p + m_k)$ forms an admissible prime constellation. Let $w(q; m_1, \dots, m_k)$ denote the number of distinct residues of $0, m_1, \dots, m_k$ modulo $q$, and let $$ C_P = 2 ^ k\prod_{\substack{q\ \text{prime} \\ q\geq 3}} \frac{1
The deck of a finite graph G is the multiset of isomorphism classes of the vertex-deleted subgraphs G − v over all vertices v. If two finite simple graphs on at least three vertices have the same deck, must they be isomorphic? Why it matters Open Problem Garden rates the problem "Outstanding". Wikipedia records that it has been verified by McKay for all graphs on at most 13 ver
We define the notion of regular primes, which are prime numbers that are coprime with the cardinality of the class group of the `p`-th cyclotomic field. We also state that there are infinitely many regular primes. Conjecture: The set of regular primes is infinite. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a7
$a(n)$ is the number of pairs of twin primes between $n^2$ and $(n+1)^2$. This counts the number of primes $p$ such that $p$ and $p+2$ are both prime, and the entire twin prime pair $(p, p+2)$ lies strictly between $n^2$ and $(n+1)^2$. That is, $n^2 < p$ and $p + 2 < (n+1)^2$. It is conjectured that $a(n)>0$ for all $n>122$. Proving this would also prove Legendre's conjecture t
Every positive integer is the sum of at most 5 tetrahedral numbers. Pollock's (tetrahedral numbers) conjecture: every integer is the sum of at most $5$ tetrahedral 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 supp
$a(n) = \gcd(\mathrm{A001008}(n), n!)$, where $\mathrm{A001008}(n)$ is the numerator of the $n$-th harmonic number $H_n = \sum_{i=1}^n \frac{1}{i}$. Conjecture: Every odd prime occurs as a term in the sequence. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal
## Kurepa's conjecture For all $n$, $$!n\not\equiv 0 \mod n$$ This appears as B44 "Sums of factorials." in [Unsolved Problems in Number Theory](https://doi.org/10.1007/978-0-387-26677-0) by *Richard K. Guy* Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal avai
If `x` is a fusible number and `y` is its successor, then the interval `[x + 1, y + 1)` can be divided into intervals `[ℓₙ, ℓₙ₊₁)`, such that the fusible numbers in `[ℓₙ, ℓₙ₊₁)` are obtained by fusing the `n + 1`st successor of `x` with a fusible number. This formalization differs from Conjecture 7.1 in the paper in four ways: (1) it is obtained from Conjecture 7.1 by plugging
Least prime $\ge n$ (version 1 of the "next prime" function). According to the "k-tuple" conjecture, $a(n)$ is the initial term of the lexicographically earliest increasing arithmetic progression of $n$ primes; the corresponding common differences are given by A061558. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634
The Lander–Parkin–Selfridge conjecture: if the sum of $n$ positive integer $k$-th powers equals the sum of $m$ positive integer $k$-th powers, with all values on the left distinct from all values on the right, then $n + m \geq k$. Formally, for positive integers $k, n, m \in \mathbb{N}$ and sequences $x : \{0, \ldots, n-1\} \to \mathbb{N}$ and $y : \{0, \ldots, m-1\} \to \mathb
The sequence $a(n)$ is the denominator of $\sum_{k=1}^n k^{\mu(k)}$, where $\mu$ is the Möbius function. Conjecture: $a(n) = \text{primorial}(n)$ for infinitely many $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 suppli
$a(0) = a(1) = 1$; $a(n) = (a(n-1) + a(n-2)) \pmod n$. All numbers appear infinitely often, i.e., for every number $k \ge 0$ and every frequency $f > 0$ there is an index $i$ such that $a(i) = k$ is the $f$-th occurrence of $k$ in the sequence. - _Klaus Brockhaus_, Aug 29 2006 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3
Product of all composite numbers between $n(n-1)/2+1$ and $n(n+1)/2$ (including boundaries), where $n(n-1)/2 = \binom{n}{2}$ and $n(n+1)/2 = \binom{n+1}{2}$. Conjecture: There are finitely many numbers such that $a(n)$ is not $\equiv 0 \pmod{a(n-1)}$. (Also mentioned in A093455.) Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision
**Four exponentials conjecture** Let $x_0, x_1$ and $y_0, y_1$ be $\mathbb Q$-linearly independent pairs of complex numbers, then some $e^{x_i y_j}$ is transcendental. 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
The sequence $a(n) = -\sum_{k=1}^n (-1)^{\lfloor (3/2)^k \rfloor}$. Is $a(n) > 0$ for all $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 de
**Firoozbakht's conjecture** The inequality $\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n}$ holds for all prime numbers $p_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 supplied for the pinned Lean 4.33.1 environment and must be
The sum of the primes in the $n$-th row of the prime number triangle: $$a(n) = \sum_{i = 1 + n(n-1)/2}^{n + n(n-1)/2} p_i$$ with $a(0) = 0$. The only positive integer $n$ such that $a(n)$ is a perfect square is $n=38$. - Carlos Eduardo Olivieri, Mar 09 2015 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380
For all odd integers $n ≥ 7$ there are prime numbers $p,q$ such that $n = p+2q$. 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
The sequence $a(n)$ is the smallest palindromic number with exactly $n$ divisors, or $0$ if no such number exists. There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural number. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11)
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