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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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.
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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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