Moser's Worm
Moser's Worm Problem What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?
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Formal Conjectures: open problems with Lean statements
Formal Conjectures is an open repository, started by Google DeepMind, of conjectures stated in Lean 4 with Mathlib. Every open problem from it that we import keeps its exact Lean statement, so a proof submitted here is checked by the Lean kernel against that statement. The collection covers Erdős problems, OEIS conjectures, Ben Green's open problems, Wikipedia's lists of unsolved problems, MathOverflow questions and more.
Source: google-deepmind/formal-conjectures. Licence: Apache License 2.0.
Moving Sofa Problem
Gerver's sofa is the unique sofa that attains the sofa constant, up to a rigid motion. The motion is needed: horizontalHallway is (-∞, 1] × [0, 1], so a leftward translate of any moving sofa is again one, obtained by sliding right and then following the original motion.
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Multiplicative order of 2 mod 2n+1
If p is an odd prime then a((p^3-1)/2) = p · a((p^2-1)/2). Because otherwise a((p^3-1)/2) < p · 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, Feb 10 2014
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No powers as partition numbers
There are no partition numbers a(k) of the form x^m, with x,m integers >1. See comment by Zhi-Wei Sun (Dec 02 2013).
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Non-Power-of-2 Almost Perfect Numbers Conjecture
Non-Power-of-2 Almost Perfect Numbers Conjecture. Does there exist an almost perfect number that is not a power of 2?
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Normality of Irrational Algebraic Numbers
The strong normality conjecture: every irrational algebraic real is absolutely normal.
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Normality of mathematical constants
π is normal in base 10.
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Number of Abelian cubes of length 3n over an alphabet of size 3
Conjecture (Peter Bala, 2022): The supercongruences a(n · p^k) ≡ a(n · p^k-1) pmodp^3k hold for the integer-indexed extension a(n) for all n ∈ ℤ ∖ 0, primes p ≥ 5, and k ≥ 1.
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Number of decompositions of 2n+1 into 2p+q, where p and q are both odd primes
Conjecture: all items for n ≥ 4 are greater than or equal to 1. This is a stronger conjecture than the Goldbach conjecture.
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Number of different products of subsets of 1, 2, …, n
Conjecture: let p ≤ 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
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Number of digits of n raised to the power of the sum of the digits of n
n=1 and 32 are two fixed points. Are there any others?
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Number of odd primes between n^2 and (n+1)^2 with (n/p) = 1
Conjecture: a(n) > 0 for all n > 0. - _Zhi-Wei Sun_, Dec 29 2012
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Number of pairs of twin primes between n^2 and (n+1)^2
It is conjectured that a(n)>0 for all n>122. Proving this would also prove Legendre's conjecture that there is a prime between n^2 and (n+1)^2. - _T. D. Noe_, Feb 28 2007
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Number of prime powers q<=n such that also q+2 is a prime power
(25,27) is the smallest pair of prime powers (q,q+2) such that both q and q+2 are not primes, conjecture: there are more (but not < 10^6).
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Number of prime powers strictly between n-th prime and (n+1)-th prime
It is conjectured that a(n) ≤ 2 for all n.
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Number of primes < n^2
Conjecture: all the numbers Σ_i=j^k 1/a(i) with 1 < j ≤ k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015
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Number of primes < n^3
Conjecture (i): for any integer k > 2, the sequence π(n^k)/n^k (n = 2, 3, …) is strictly decreasing, where π(x) denotes the number of primes not exceeding x. - Zhi-Wei Sun, Oct 17 2015
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Number of primes in n-th row of triangle k^2 - k + p_n
Conjecture: a(n) < n for n > 13.
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Number of primes p such that 2^n ≤ p ≤ 2^n + prime(n)
For any n > 0, is there always at least one prime p such that 2^n ≤ p ≤ 2^n + prime(n)? (checked up to n = 250).
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Number of primes p such that n^n ≤ p ≤ n^n + n^2
Question: for any n > 0, is there at least one prime p such that n^n ≤ p ≤ n^n + n^2? In this case, that would be stronger than the Schinzel conjecture: "for m > 1 there's at least one prime p such that m ≤ p ≤ m + log(m)^2" since n^2 < log(n^n)^2 = n^2 log(n)^2.
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Number of refactorable numbers (A033950) ≤ 10^n
Colton's conjecture [Co99] as stated by Zelinsky [Ze02]: for every n, the number of refactorable numbers ≤ n is at least half the number of primes ≤ n, i.e. π(n) ≤ 2 T(n).
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Number of squares bmod n
n^2 ≡ 1 pmoda(n)(a(n)-1) if and only if n is an odd prime. - Thomas Ordowski, Jun 08 2017
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Number of symbols '*' and '^' to write the canonical prime factorization of n
It is conjectured that 1,2,3,4,5,6,7,9,11 are the only positive integers which cannot be represented as the sum of two elements of indices n such that a(n) = 1.
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Number of times n^2 + s^2 is prime for positive integers s < n
Conjecture: a(n) > 0 for all n > 1.
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Number of ways to express n as sum of square, pentagonal, and hexagonal numbers
In April 2009, _Zhi-Wei Sun_ conjectured that a(n) > 0 for every n = 0, 1, 2, 3, ….
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Number of zeros in ternary representation of 2^n
Conjecture from N. J. A. Sloane: a(n) > 0 for n > 15.
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Numbers n such that n^2 + π(n) is prime
Conjecture: the sequence A228828 is infinite.
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Numbers n such that the perfect deficiency of n is ≤ 10.
Is 1155 the last odd number in this sequence? (1155 is the 59th term starting from 1, corresponding to a(58) = 1155).
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Numbers n such that φ(n) = φ(n + φ(n))
Conjecture: Except for the first term all terms are even.
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Numerator of 1/det(M) for M[i,j] = 1/lcm(i,j)
"Conjecture: 1/det(M) is an integer only for n: 1 to 34, 36 and 38." - _Robert G. Wilson v_, Aug 02 2015
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Numerator of a sum involving binomial coefficients
We conjecture that u(p-1) == 0 (mod p^4) for all primes p, with a finite number of exceptions that depend on m.
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Numerator of ζ(4n)/ζ(2n)^2 (with a(0)=2 instead of -2)
Conjecture: if an integer n > 1 is odd, then ζ(2n)/ζ(n)^2 is irrational. Cf. W. Kohnen (link) and my conjecture in A348829. - Thomas Ordowski, Jan 05 2022
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Numerator of Σ_k=1^n 2^k/k.
Conjecture: for n > 3, textrmnumerator(-2/n + Σ_k=1^n 2^k/k) == 0 (textrmmod n^2) if and only if n is prime.
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Odious primes minus evil primes among first n primes
Shevelev conjectures that a(n) ≥ 0 for n > 3.
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Open Quantum Problem 13: Mutually unbiased bases
Special case in dimension 6: determine the maximal number of mutually unbiased orthonormal bases in ℂ^6.
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Open Quantum Problem 23: SIC-POVMs
Benchmark open subproblem: existence of a SIC-POVM in dimension 56.
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Open Quantum Problem 35: existence of absolutely maximally entangled pure states
Open benchmark statement: does an AME(8,4) state exist?
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Open questions about Fermat numbers
Are Fermat numbers composite for all n > 4?
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Open questions on irrationality of numbers
Are e and π algebraically independent?
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Open questions on transcendence of numbers
e + π is transcendental.
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Oppermann's Conjecture
For every integer x ≥ 2 there exists a prime between x(x-1) and x^2.
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Packing
What is the smallest square that can contain 11 unit squares? Reference: Wikipedia
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Partial sums of C(2n, n)^2
Conjecture: For any positive integer n, the polynomials Sum_k=0^n binomial(2k,k)^2x^k and Sum_k=0^n binomial(2k,k)^2x^k/(k+1) are irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 23 2013
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Particular values of the Riemann zeta function
ζ(5) is irrational.
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Pebbling number conjecture
The pebbling number conjecture: the pebbling number of a Cartesian product of connected graphs is at most equal to the product of the pebbling numbers of the factors. See Asplund, Hurlbert, and Kenter.
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Pfister's problem on the Pythagoras number of ℝ(X_1, …, X_n)
Pfister's problem (Problem 1 of [Pfister1971, §4]): what is the true value of p(ℝ(X_1, …, X_n)), as a function of n?
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Pierce–Birkhoff conjecture
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).
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Pierpont primes
There are infinitely many Pierpont primes.
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Pollock's (tetrahedral numbers) conjecture
Pollock's (tetrahedral numbers) conjecture: every integer is the sum of at most 5 tetrahedral numbers.
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Polynomial-time computability of factoring
The integer factorization problem: Can the prime factorization of a positive integer be computed in polynomial time? We state the problem by asking if Nat.primeFactorsList is polynomial-time computable (assuming typical encodings of ℕ and List ℕ into bitstrings). Reference: Wikipedia
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Practical numbers
Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical number. - Hal M. Switkay, Jan 28 2023
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Prime Triplet Conjecture
Are there infinitely many tuples of three consecutive primes (p, q, r) such that r - p = 6?
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Prime Tuples Conjecture
For any k ≥ 2, let a₁,...,aₖ and b₁,...,bₖ be integers with aᵢ > 0. Suppose that for every prime p there exists an integer n such that p ∤ ∏ i, (aᵢ n + bᵢ). Then there exist infinitely many n such that aᵢ n + bᵢ is prime for all i.
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Prime-th recurrence with reversal at each step
Starting at a positive value other than a(0) = 1, does this sequence ever go into a loop? The positivity hypothesis is required because the source recurrence uses the one-based prime index p₁ = 2; the x = 0 branch above is only an artifact of making aStartAt total on ℕ.
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Primes and perfect squares
Are there infinitely many primes p such that p - 1 is a perfect square? In other words: Are there infinitely many primes of the form n^2 + 1?
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Primes of the form 2^n + 2^i + 1
Conjecture (A81091): There are infinite primes of the form 2^n + 2^i + 1, with 0 < i < n.
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Primitive roots of the form k² + 1
Zhi-Wei Sun's Conjecture (A239957): Every prime p has a primitive root 0 < g < p of the form k^2 + 1, where k is an integer.
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Product of two consecutive primes modulo the next prime
Conjecture: For x > 10^9, the most frequent value in a(n), n=1… x, has form 120k.
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Quasiperfect Numbers
Quasiperfect Numbers Conjecture. Do quasiperfect numbers exist?
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Ramanujan τ-function
Lehmer's conjecture: τ(n) ≠ 0 for all n > 0.
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Ramsey numbers
The open problem: determine the Ramsey number R(5,5). It is known that 43 ≤ R(5,5) ≤ 46.
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Rational distance problem
Does there exist a point in the plane at rational distance from all four vertices of the unit square?
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Recurrence a(n) = (a(n-1) + a(n-2)) pmod n
All numbers appear infinitely often, i.e., for every number k ≥ 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
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Recurrence involving LCM: a(n) = x(n+1)/x(n) - 2
Conjecture: For prime p such that p-2 is not a prime, a(p-1) = p. - _Bill McEachen_, Sep 26 2025
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Recurrence with bitwise XOR
Conjecture: the sequence contains 8 zeros.
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Reed's omega, delta, and chi conjecture
For a graph G, we define Δ(G) to be the maximum degree, ω(G) to be the size of the largest clique subgraph, and χ(G) to be the chromatic number. Reed's omega, delta, and chi conjecture states that χ(G) ≤ ⌈ 1/2(ω(G) + Δ(G) + 1) ⌉.
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Representations as p + 2^x + 11 · 2^y with p ≡ 1 pmod 6
On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n) = 0 if and only if n < 16 or n ∈ 18, 21, 24, 51, 84, 1011, 59586; in other words, except for 35, 41, 47, 101, 167, 2021, 119171, any odd integer greater than 30 can be written as the sum of a prime congruent to 1 bmod 6, a positive power of 2 and…
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Representations with prime conditions
Zhi-Wei Sun's Conjecture (A232174): Any integer n > 1 can be written as x + y with x, y > 0 such that both x + ny and x^2 + ny^2 are prime.
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Resolution of singularities
Resolution of singularities in positive characteristic. Let k be a perfect field of characteristic p > 0 and let X be an integral scheme that is separated and of finite type over k. Then there is an integral scheme Y that is smooth over k together with a proper birational morphism Y → X.
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Riesel Problem
It is conjectured that the integer k = 509203 is the smallest Riesel number, that is, the first n such that a(n) = -1 is 254602.
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Ringel's Conjecture
For any tree T with n edges, the complete graph K_2n+1 decomposes into 2n+1 edge-disjoint copies of T. A "copy" of T is the image T.map(f_i) of T under a vertex embedding f_i : V hookrightarrow Fin(2n+1); the copies are pairwise edge-disjoint and together cover every edge of K_2n+1.
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Rowland-style prime-generating recurrence
Conjecture: Every record of differences a(n)-a(n-1) more than 5 is the greater of twin primes (A006512).
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Rudin's conjecture on squares in arithmetic progressions
Rudin's conjecture. The maximal number of squares among the first N terms of a non-trivial arithmetic progression grows at most like √(N): Q(N) = O(√(N)).
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Schanuel's Conjecture
Given any set of n complex numbers z_1, ..., z_n that are linearly independent over ℚ, the field extension ℚ(z_1, ..., z_n, e^z_1, ..., e^z_n) has transcendence degree at least n over ℚ.
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Scholz conjecture on addition chains
The Scholz conjecture, also known as the Scholz-Brauer conjecture, asserts that for every positive integer n, the addition-chain length of 2^n - 1 is at most n - 1 + ℓ(n).
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Selfridge's conjectures
PSW conjecture (Selfridge's test) Let p be an odd number, with p ≡ ± 2 pmod5, 2^p-1 ≡ 1 pmodp and F_p+1 ≡ 0 pmodp, then p is a prime number.
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Serre's multiplicity conjectures
Positivity conjecture. Let R be a regular local ring and let M, N be finitely generated R-modules such that M otimes_R N has finite length. If dim M + dim N = dim R, then χ(M, N) > 0. The hypothesis on dimensions forces M and N to be nonzero, since the dimension of the zero module is bot.
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Serre's uniformity conjecture over the rationals
Serre's uniformity question over ℚ [Ser72, Lem17]: is there a bound C such that every non-CM elliptic curve over ℚ has surjective mod-p Galois representation for every prime p > C?
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Sidorenko's conjecture (1993)
Sidorenko's conjecture (1993). For every finite bipartite simple graph H and every finite simple graph G: t(H, G) ≥ t(K_2, G)^e(H), where K_2 denotes the single-edge graph on 2 vertices (i.e. completeGraph (Fin 2)).
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Sierpiński number
The Sierpiński problem (Selfridge's conjecture). Is 78557 the smallest Sierpiński number? Selfridge conjectured that 78557 is the smallest Sierpiński number.
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Singmaster's conjecture
Singmaster's conjecture: the number of times any number t > 1 appears in Pascal's triangle is bounded.
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Smallest composite c such that textrmprimorial(n) + c is prime
Conjecture: liminf_n → ∞ a(n)/p_n+1^2 = 1 < limsup_n → ∞ a(n)/p_n+1^2 = 2. - Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015
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Smallest factorial containing exactly n 6's
It is conjectured that a(24) = 0 since no factorial less than 10000 contained just 24 sixes.
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Smallest index k > n such that (p_k+p_k+1)/(p_n+p_n+1) is an integer ≥ 2
Conjecture: the sequence is infinite, that is, for every n ≥ 1 there is some k > n with S(n) | S(k), so that a(n) is defined.
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Smallest m > 0 such that n · 2^m + 1 is prime
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
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Smallest m > 0 such that there are no primes between nm and n(m+1) inclusive.
Sierpinski's conjecture (1958) is precisely that a(n) >= n for all n.
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Smallest m such that n^3 + m^3 + 1 is prime
Conjecture 2: For any k ≥ 3, there are infinitely many primes of the form n^k + m^k + 1 for n, m ≥ 1. - _Ulrich Krug_, 2009
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Smallest number k such that kn + 1 is prime
Conjecture: for every n > 1 there exists a number k < n such that nk + 1 is a prime.
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Smallest number m such that 2^n - m and 2^n + m are primes
Conjecture: a(n) = O(n^3). The source defines a(n) as the least m with 2^n - m and 2^n + m prime, so it implicitly asserts that such an m exists. Since a n = 0 when no such m exists, the existence of a prime pair is stated explicitly for all sufficiently large n.
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Smallest palindrome with exactly n divisors
There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural number.
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Smallest prime ≡ 1 pmod n
"Conjecture: a(n) < n^2 for n > 1. - _Thomas Ordowski_, Dec 19 2016"
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Smallest prime formed by concatenation n, n-1, …, n-k
Conjecture: There are infinitely many composite numbers n such that a(n) is nonzero.
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Smallest r such that (concatenation of n, r times) · 10 + 1 is prime
What is the smallest integer m > 1 such that a(10^m) is nonzero? - _Farideh Firoozbakht_, Jan 07 2015
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Smallest x such that σ(x) bmod x = n
At present, the 0 entry for n = 5 is only a conjecture. That is, it is conjectured that there is no positive integer x such that σ_1(x) bmod x = 5.
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Solitary Numbers
Is 10 a solitary number? The smallest positive integer whose solitary status is currently unresolved is 10, with abundancy index σ(10) / 10 = 9/5.
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Some conjectures about ranks of elliptic curves over ℚ
Conjecture by Goldfeld and Katz–Sarnak: if elliptic curves over ℚ are ordered by their heights, then 50% of the curves have rank 0 and 50% have rank 1. See p. 28 of https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Bhargava.pdf.
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Sparse Ruler
Wichmann's conjecture on optimal rulers. Every optimal ruler of sufficiently large length is a Wichmann ruler W(r, s) (up to reflection, i.e. reversing the segment list). Posed by Wichmann [Wi63].
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Spectral sets and weak tiling
[KLM2023, Problem 7.1] asks whether a bounded, measurable, nowhere dense subset Ω ⊂ ℝ^d of positive measure can be spectral. The answer is known to be negative for d = 1, so the dimension is restricted to d ≥ 2, where the problem is open.
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Steiner Systems
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 systems must exist for sufficiently large n. Reference: Large Steiner Systems
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Strong Sensitivity Conjecture (bs(f) ≤ s(f)^2)
Strong Sensitivity Conjecture, for every Boolean function f : 0,1^n → 0,1, bs(f) ≤ s(f)^2. We call this the strong sensitivity conjecture because the original sensitivity conjecture only asked for a polynomial bound in terms of s(f).
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