google-deepmind/formal-conjectures

What is the conjecture The Ramsey number $R(5,5)$ is defined as the smallest positive integer $n$ such that every 2-coloring of the edges of the complete graph $K_n$ contains either a monochromatic clique of 5 vertices or a monochromatic independent set of 5 vertices. Equivalently, for any graph $G$ on $n$ vertices, either $G$ or its complement $\overline{G}$ contains a clique of size 5. The exact value of $R(5,5)$ remains unknown, but it is bounded: $43 \leq R(5,5) \leq 48$. Also add conjectures for other small values. (This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.) Sources: https://oeis.org/A212954, https://mathworld.wolfram.com/RamseyNumber.html, https://gilkalai.wordpress.com/2017/03/29/r55-%E2%89%A4-48/, https://ajc.maths.uq.edu.au/pdf/5/ocr-ajc-v5-p13.pdf, https://math.mit.edu/~apost/courses/18.204_2018/ramsey-numbers.pdf Prerequisites needed Formalizability Rating: 1/5 (0 is best) (as of 2026-02-19) Bui

What is the conjecture The Van der Waerden number $W(r, k)$ is the smallest positive integer $n$ such that for every partition of the set ${1, 2, \ldots, n}$ into $r$ disjoint subsets, at least one subset contains an arithmetic progression of length $k$. The core open problem is to determine exact values of $W(r, k)$ for arbitrary pairs $(r, k)$. Known exact values are extremely rare: only the pairs $(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (4, 3), (5, 3), (6, 3), (2, 8), (3, 4)$ have been computed exactly. Computing even the next value remains an open computational challenge. Related open problems include improving upper and lower bounds for $W(r, k)$ and determining whether the mixed van der Waerden numbers are concave. (This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.) Sources: Wikipedia: Van der Waerden number Wolfram MathWorld: Van der Waerden Numbers Open Problem Garden: Concavity of van d

What is the conjecture Let $a_n$ denote the number of distinct flat foldings of a $1 \times n$ rectangular strip with horizontal and vertical creases marked as mountain or valley folds. The sequence ${a_n}$ (OEIS A000136) begins: $$a_1 = 1, \quad a_2 = 2, \quad a_3 = 6, \quad a_4 = 16, \quad a_5 = 50, \quad a_6 = 144, \quad a_7 = 462, \quad a_8 = 1392, \quad \ldots$$ The Growth Rate Conjecture: Determine the precise limiting ratio: $$\lim_{n \to \infty} \frac{a_{n+1}}{a_n}$$ It is known that this limit exists and lies in the interval $[3.3868, 3.9821]$, indicating exponential growth of $a_n$. However: The exact value of the limit remains unknown. It is unclear whether the limit has a closed form or is a transcendental number. Whether this limit can be expressed in terms of known mathematical constants is an open question. The similar problem for $2 \times n$ maps (OEIS A001415) also lacks a known growth rate formula. (This description may contain subtle errors especially on more comp

What is the conjecture For a positive integer $n$, define the aliquot sequence as the sequence $(s_0, s_1, s_2, \ldots)$ where $s_0 = n$ and $s_{i+1} = \sigma(s_i) - s_i$ for $i \geq 0$. Here $\sigma(m)$ denotes the sum of all positive divisors of $m$, and $\sigma(m) - m$ is the sum of proper divisors of $m$. The Catalan-Dickson conjecture states that every aliquot sequence is bounded. Specifically, every such sequence either terminates (reaches 1), reaches a perfect number (a number equal to the sum of its proper divisors), or enters a cycle of sociable numbers (numbers that are part of a periodic cycle under the aliquot iteration, including amicable pairs as special cases of length 2). (This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.) Sources: Wikipedia: Aliquot sequence Wolfram MathWorld: Catalan's Aliquot Sequence Conjecture ProofWiki: Catalan-Dickson Conjecture OEIS: Aliquot sequences Carl Pomerance: Ali

What is the conjecture A periodic group is a group in which every element has finite order. A finitely presented group is a group that can be described using a finite set of generators and finitely many relations between them. The conjecture asks: If $G$ is a finitely presented group and $G$ is periodic, must $G$ be finite? Equivalently: Does there exist a finitely presented group $G$ such that $G$ is periodic but $G$ is infinite? (This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.) Sources: https://en.wikipedia.org/wiki/Periodic_group, https://en.wikipedia.org/wiki/Presentation_of_a_group, Olshanskii, A. Yu. (1991). Geometry of defining relations in groups (Mathematics and Its Applications), Hall, M. (1959). The theory of groups (Macmillan) Prerequisites needed Formalizability Rating: 1.5/5 (0 is best) (as of 2026-02-03) Building blocks (1-3; from search results): Group and finite order concepts (via orderOf in M