Partitions n = p + q with p, 3p - 10, and (p - 1)q - 1 prime
Conjecture: a(n) > 0 for all n > 5.
No claims yet Be the first →
Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
All fields Mathematics 1261Computer science 31Physics & astronomy 22Chemistry & materials 13Life sciences 6Earth & society 6
1339 shown· page 24 of 27
Partitions n = x + y with x^2 + xy + y^2 prime
Conjecture: a(n) > 0 for all n > 1 with the only exception n = 8.
No claims yet Be the first →
Partitions of n into x + y with (xy)^2 + 1 prime
Conjecture: a(n) > 0 if n is different from 1, 6, 16, 24.
No claims yet Be the first →
Partitions of n into x + y with (xy)^2 + xy + 1 prime
Conjecture: a(n) > 0 for all n > 1.
No claims yet Be the first →
Partitions of n into x + y with x(x+1)/2 + y^2 prime
Conjecture: a(n) > 0 for all n > 1.
No claims yet Be the first →
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.
No claims yet Be the first →
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?
No claims yet Be the first →
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).
No claims yet Be the first →
Pierpont primes
There are infinitely many Pierpont primes.
No claims yet Be the first →
Pittel's second-moment conjecture for nearly balanced markets
Pittel conjectures that if d_n ≥ 1 and d_n/n → 0, then the second moment of the number S_n of stable matchings is O((E[S_n])^2). The bound may depend on the sequence d_n.
No claims yet Be the first →
Pittel's stable-matching hyperbola threshold conjecture
Pittel conjectures a threshold p_(n) for the rank pairs across stable matchings to trace out a hyperbola. For all sufficiently large n, it lies between 9/4(log n)^2/n and 1. The fixed reciprocal arc (t,t^-1) for t ∈ [2/3,3/2] specifies the curve to cover in this formulation.
No claims yet Be the first →
Pollock's (tetrahedral numbers) conjecture
Pollock's (tetrahedral numbers) conjecture: every integer is the sum of at most 5 tetrahedral numbers.
No claims yet Be the first →
Polynomial-time computability of comparisons of sums of square roots
Is the square-root sum problem polynomial time (single sum version)? Reference: Wikipedia
No claims yet Be the first →
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
No claims yet Be the first →
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
No claims yet Be the first →
Prime quadruplets generated by p = 3φ(k) + φ(n-k) - 1
Conjecture: a(n) > 0 for all n > 1075.
No claims yet Be the first →
Prime Triplet Conjecture
Are there infinitely many tuples of three consecutive primes (p, q, r) such that r - p = 6?
No claims yet Be the first →
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.
No claims yet Be the first →
Prime values in recurrence a(n+1) = |a(n) - gcd(a(n), 7n+6)|
It is conjectured that for all n > 2, a(n) = 0 implies that 7n+6 = a(n+1) is prime, cf. A186259.
No claims yet Be the first →
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.
No claims yet Be the first →
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?
No claims yet Be the first →
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.
No claims yet Be the first →
Primes of the form 3^n ± 3^k ± 1
"Conjecture: all elements of this sequence are greater than 0."
No claims yet Be the first →
Primes p < n with n^2 - n + p and n^2 + n - p prime
Conjecture: a(n) > 0 for all n > 2732.
No claims yet Be the first →
Primes p such that n - (p n' - 1) and n + (p n' - 1) are both prime
Conjecture: a(n) > 0 for all n > 6.
No claims yet Be the first →
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.
No claims yet Be the first →
Primorial squared plus one
It is unknown whether or not numbers in this sequence are always squarefree. - _John M. Campbell_, Apr 21 2011
No claims yet Be the first →
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.
No claims yet Be the first →
Pudlák's conjectures on incompleteness in the finite domain
SAT has no p-optimal proof system (Pudlák's conjecture SAT [Pu17]). No proof system for SAT p-simulates every other proof system for SAT.
No claims yet Be the first →
Quasiperfect Numbers
Quasiperfect Numbers Conjecture. Do quasiperfect numbers exist?
No claims yet Be the first →
Ramanujan τ-function
Lehmer's conjecture: τ(n) ≠ 0 for all n > 0.
No claims yet Be the first →
Ramsey numbers
The open problem: determine the Ramsey number R(5,5). It is known that 43 ≤ R(5,5) ≤ 46.
No claims yet Be the first →
Rational distance problem
Does there exist a point in the plane at rational distance from all four vertices of the unit square?
No claims yet Be the first →
Recamán's sequence
“I conjecture that every number eventually appears” — N. J. A. Sloane, OEIS A005132. Sloane later added that he was no longer sure every number appears. The coverage question remains open and is formalized here as surjectivity onto the natural numbers.
No claims yet Be the first →
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
No claims yet Be the first →
Recurrence a(n+1) = 2n(n+1)a(n) - n^4 a(n-1) and associated polynomials
Conjecture, (i): S_n(x) is irreducible over the field of rational numbers for every n = 1, 2, 3, … - _Zhi-Wei Sun_, Mar 20 2013
No claims yet Be the first →
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
No claims yet Be the first →
Recurrence with bitwise XOR
Conjecture: the sequence contains 8 zeros.
No claims yet Be the first →
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) ⌉.
No claims yet Be the first →
Refining Lagrange's four-square theorem with (5x^2 + 7y^2 + 9z^2)yz a square
Conjecture (i): a(n) > 0 for all n > 0.
No claims yet Be the first →
Remainder of 2^n+2 modulo n
"Conjecture: every integer k ≥ 0 appears in a(n) at least once."
No claims yet Be the first →
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…
No claims yet Be the first →
Representations n = k^2 + m (0 < k^2 ≤ m) with σ(k^2) + φ(m) prime
Conjecture (i): a(n) > 0 for all n > 1. - _Zhi-Wei Sun_
No claims yet Be the first →
Representations of n = p + q + (n bmod 2)q with Jacobi symbol constraints
Conjecture: a(n) > 0 for all n > 5. - _Zhi-Wei Sun_, Dec 30 2012
No claims yet Be the first →
Representations of n as a sum of generalized heptagonal, octagonal, and nonagonal numbers
Conjecture: a(n) > 0 for all n. - _Zhi-Wei Sun_
No claims yet Be the first →
Representations of n as the sum of two pentagonal and a generalized decagonal number
Conjecture: a(n) > 0 for all n. - _Zhi-Wei Sun_
No claims yet Be the first →
Representations of n as x^2 + φ(y^2) + φ(z^2)
Conjecture (i): a(n) > 0 for all n > 6.
No claims yet Be the first →
Representations of n using prime pairs with prime gaps of 6
Conjecture: a(n) > 0 for all even n > 8012 and odd n > 15727.
No claims yet Be the first →
Representations using Sophie Germain primes and twin prime pairs
Conjecture (i), first part: a(n) > 0 for all n > 1.
No claims yet Be the first →
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.
No claims yet Be the first →
Browse by field
CombinatoricsNumber theoryGraph theoryAlgebraGeometryAnalysisProbabilityLogic & formalisationAlgorithmsComplexityComputabilityOptimisationCryptography theoryMachine learningPhysicsTheoretical physicsAstrophysics & cosmologyQuantum informationChemistryMaterialsBiologyNeuroscienceEcologyClimateEarth scienceEconomics