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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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 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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