Number of 0 < k < n-2 with p = φ(k) + φ(n-k)/2 + 1 and p_p - p ± 1 prime
Conjecture: a(n) > 0 for all n > 31.
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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.
Number of 0 < k < n/2 such that φ(k)φ(n-k) is a square
Conjecture: (i) a(n) > 0 for all n > 8.
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Number of 0's in the binary expansion of n^2
Conjecture: For every x ≥ 0, there is an i such that a(n) > x for all n > i.
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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 integers 0 < k < n such that (k + 1)^φ(n - k) + k is prime
Conjecture (i): a(n) > 0 for all n > 1.
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Number of integers 0 < k < n such that k φ(n - k) + 1 is a square
Conjecture (i): a(n) > 0 if n is not a divisor of 6.
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Number of integers 0 < k < n such that p = k + φ(n - k) and 2(n - p) + 1 are both prime
Conjecture: a(n) > 0 for all n > 2.
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Number of integers 0 < k ≤ n such that π(π(k n)) is a square
Conjecture (i): a(n) > 0 for all n > 0.
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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 positive integers k < n such that n^2 - π(k n) is prime
Conjecture (i): a(n) > 0 for all n > 1.
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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 between n^2 and n^2 + (log_2 n)^2
Conjecture: For every n > 4765516, a(n) > 0.
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Number of primes between n^3 - n and n^3
Conjecture: For every n > 13, a(n) > 0.
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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 of the form 2^n ± 2^k ± 1 with 0 ≤ k < n
Conjecture: all terms of this sequence are greater than 0. - _Lei Zhou_, Oct 05 2011
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Number of primes p < n such that p φ(n - p) - 1 is also prime
Conjecture: a(n) > 0 for all n > 3.
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Number of primes p < n with prime(p)^2 + (prime(n) - 1)^2 prime
Conjecture (i): a(n) > 0 unless n divides 6.
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Number of primes p ≤ (n+1)/2 with ⌊√(n-p)⌋ prime
Conjecture (i): a(n) > 0 for all n > 5.
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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 ways to write n = p + q (q > 0) with p and (φ(p)φ(q))^4 + 1 prime
Conjecture (i): a(n) > 0 for all n > 2.
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Number of ways to write n = x + y (0 < x < y < n) with 6x ± 1 and 6y ± 1 prime
Conjecture: If n > 200 is not among 211, 226, 541, 701, then a(n) > 0. - _Zhi-Wei Sun_, Jan 03 2013
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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 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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Odd part of the digital sum of 3^n divided by the maximal power of 3
Conjecture: Consider the sequence b(n) such that b(1)=2, b(2)=3, and for n ≥ 3, b(n)=a(n-2). If we apply the Eratosthenes-like sieve to b(n) and remove 1s, then we obtain a sequence of primes. - _Vladimir Shevelev_, Dec 12 2014
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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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Ordered ways to write 2n = p + q with p, q and p_p+2 + 2 all prime
Conjecture: (i) a(n) > 0 for all n > 2.
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Ordered ways to write n = k + m with p_k + 4 and p_p_m + 4 both prime
Conjecture: For each d = 1, 2, 3, … there is a positive integer N(d) for which any integer n > N(d) can be written as k + m with k > 0 and m > 0 such that p_k + 2d and p_p_m + 2d are both prime.
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Ore's harmonic numbers
"Ore conjectured that 1 is the only odd harmonic number." Reference: A001599
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Packing
What is the smallest square that can contain 11 unit squares? Reference: Wikipedia
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Pairs j, k with 0 < j < k ≤ n, k ≡ 1 pmod j, and π(jn) | π(kn)
Conjecture: a(n) > 0 for all n > 1.
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Parities of partition numbers
The natural density of n where the partition number p(n) is even (resp. odd) exists and equals 1/2.
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Partial sums of (-1)^k - Ω(k)
"On May 16 2012, Zhi-Wei Sun conjectured that a(n) is positive for each n > 4." - _Zhi-Wei Sun_, May 19 2012
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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(7) is irrational.
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Partitions 2n + 1 = x + y with x^4 + y^4 prime
Conjecture: a(n) > 0 for all n ≥ 1.
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Partitions n = p + q with p, 3p - 10, and (p - 1)q - 1 prime
Conjecture: a(n) > 0 for all n > 5.
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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.
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Partitions of n into x + y with (xy)^2 + 1 prime
Conjecture: a(n) > 0 if n is different from 1, 6, 16, 24.
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Partitions of n into x + y with (xy)^2 + xy + 1 prime
Conjecture: a(n) > 0 for all n > 1.
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Partitions of n into x + y with x(x+1)/2 + y^2 prime
Conjecture: a(n) > 0 for all n > 1.
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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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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.
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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.
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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 comparisons of sums of square roots
Is the square-root sum problem polynomial time (single sum version)? Reference: Wikipedia
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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 quadruplets generated by p = 3φ(k) + φ(n-k) - 1
Conjecture: a(n) > 0 for all n > 1075.
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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 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.
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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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Primes of the form 3^n ± 3^k ± 1
"Conjecture: all elements of this sequence are greater than 0."
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Primes p < n with n^2 - n + p and n^2 + n - p prime
Conjecture: a(n) > 0 for all n > 2732.
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Primes p such that n - (p n' - 1) and n + (p n' - 1) are both prime
Conjecture: a(n) > 0 for all n > 6.
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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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Primorial squared plus one
It is unknown whether or not numbers in this sequence are always squarefree. - _John M. Campbell_, Apr 21 2011
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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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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.
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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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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.
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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 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
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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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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.
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