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Conjecture: $f(n, k) = 2$ for infinitely many cases, where $k = a(n)$. We assume $a(n) \ne 0$ (i.e., that a suitable $k > n$ always exists), as `sInf` evaluates to $0$ on an empty set. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop defini
The sequence is defined by $a(1) = 2$ and for $n \ge 2$: $a(n) = a(n-1) + \gcd(n, a(n-1))$ if $n$ is even, and $a(n) = a(n-1) + \gcd(n-2, a(n-1))$ if $n$ is odd. Conjecture: Every record of differences $a(n)-a(n-1)$ more than 5 is the greater of twin primes (A006512). Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db46347
The sequence is defined by $$a(n) = \sum_{k=0}^n (-4)^k \binom{n}{k}^2 \binom{n-k}{k}^2.$$ If $p$ is a prime with $p \equiv 1, 9 \pmod{20}$ and $p = x^2 + 5y^2$ with $x, y$ integers, then $\sum_{k=0}^{p-1} a(k) \equiv 4x^2 - 2p \pmod{p^2}$. - _Zhi-Wei Sun_, Jul 01 2010 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634
In this problem, a function $h : \mathbb{N} \to\mathbb{N}$ is defined maximally by some counting property. The problem asks to estimate $h(n)$. This has been interpreted here as asking for $\Theta(h(n))$. The principal version includes `answer(sorry)` for an unknown function. On the other hand, the best known upper bound is $\sqrt{n}$ and the best known lower bound is $(n\log(n
Erdős and Sós conjectured that $f_k(N)\sim \frac{1}{2}\left(1+\sum_{1\leq r\leq k-2}\frac{1}{4^r}\right) N$, where $f_k(N)$ is the minimal size of a subset of $\{1, \dots, N\}$ guaranteeing $k$ elements have all pairwise sums in the set. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (c
$F(n) / \log n \to \infty as n \to \infty$ 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 prov
Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + C n^{\frac{1}{4}})$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal availability A Prop def
Primes $p$ such that $p+2$ is also prime. Primes $p_k$ such that $p_k! \equiv 1 \pmod{p_{k+1}}$ with the exception of $p_{991} = 7841$ and other unknown primes $p_k$ for which $(p_k+1)(p_k+2)\cdots(p_{k+1}-2) \equiv 1 \pmod{p_{k+1}}$ where $p_{k+1} - p_k > 2$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2
$a(n) = x(n+1)/x(n) - 2$ where $x(1)=1$ and $x(n) = 2 x(n-1) + \operatorname{lcm}(x(n-1),n)$ for $n > 1$. Conjecture: For prime $p$ such that $p-2$ is not a prime, $a(p-1) = p$. - _Bill McEachen_, Sep 26 2025 Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal av
The sequence is defined as $$a(n) = \mathrm{round}\left(\frac{\mathrm{round}(\sqrt{n!})}{\left|(\mathrm{round}(\sqrt{n!}))^2 - n!\right|}\right)$$ for $n \ge 2$. This sequence suggests that the distance between a factorial and the closest power is tightly bounded. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a7
Show that for any rational $\alpha \in [1,2)$ there exists a bipartite graph $G$ such that $$\mathrm{ex}(n;G)\asymp n^{\alpha}.$$ 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
Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$? 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 accepte
$a(n) = p_{4n-3}p_{4n} - p_{4n-2}p_{4n-1}$ where $p_k$ is the k-th prime number (1-indexed). This suggests the ratio is approaching a limit close to 0.87. Formalized as: The sequence of ratios $P(N)/Neg(N)$ converges to a limit L, and L is in the interval (0.8, 0.9). Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db463473
$a(n)$ is the number of primes in the $n$-th row of the triangle $T(n, k) = k^2 - k + p_n$ for $1 \le k \le n$, where $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \dots$). Conjecture: $a(n) < n$ for $n > 13$. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11). Formal avail
Show that for $k\geq 3$ $$\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}.$$ This problem is #46 in Extremal Graph Theory in the graphs problem collection. 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 Le
Let $r\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges of the induced $K_{r+1}$. In other words, there is no balanced colouring. A conjecture of Erdős and Gyárfás [ErGy99]. Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f8
Define $F(n) = 2^{2^n} + 1$ (the $n$-th Fermat number) and $M(n) = 2^n - 1$ (the $n$-th Mersenne number). Then $a(n) = F(n) + M(n) - 1 = 2^{2^n} + 2^n - 1$. The first 5 entries are primes. Are there infinitely many primes in this sequence? Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0
Let $A \subset \mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c \in A$ (aside from the trivial coincidences). Is it true that $$\liminf_{N \to \infty} \frac{\lvert A \cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?$$ Mathematical status Open: marked `research open` in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77b
The **Odd Noncototient Conjecture**: every non-cototient is even. Equivalently, every odd natural number is of the form $n - \phi(n)$ for some $n$. This is the unconditional form of `erdos_418.variants.conditional`, which derives the odd case from a strengthening of the Goldbach conjecture. See [Wikipedia: Noncototient]. Mathematical status Open: marked `research open` in googl
Conjecture: $a(n) = O(n^3)$. 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