Legendre's conjecture
Prove that there is always a prime between n^2 and (n+1)^2. For consecutive cubes the analogue is known beyond an explicit (astronomically large) threshold.
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Open problems about prime numbers
Questions about how primes are distributed: gaps between consecutive primes, primes in special forms and sequences, and sums and products involving primes. Most come from Erdős's problem lists; many have a finite search that can refute them or collect evidence.
The (binary) Goldbach conjecture
Prove that every even integer greater than 2 is the sum of two primes. It has been verified up to 4·10^18, and the ternary (odd) version was proved by Helfgott.
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The twin prime conjecture and bounded prime gaps
Prove that there are infinitely many primes p with p + 2 prime. Intermediate target is to lower H_1 = liminf (p_{n+1} − p_n), proved at most 246 in published work; unrefereed 2026 work claims 240, 212 and 186.
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Erdős Problem #10
Is there some k such that every large integer is the sum of a prime and at most k powers of 2?
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Erdős Problem #1055
A prime p is in class 1 if the only prime divisors of p+1 are 2 or 3. In general, a prime p is in class r if every prime factor of p+1 is in some class ≤ r-1, with equality for at least one prime factor. Are there infinitely many primes in each class?
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Erdős Problem #1059
Are there infinitely many primes p such that p - k! is composite for each k such that 1 ≤ k! < p?
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Erdős Problem #1137
Let d_n=p_n+1-p_n, where p_n denotes the nth prime. Is it true that max_n < xd_nd_n-1/(max_n < xd_n)^2→ 0 as x→ ∞?
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Erdős Problem #1139
Let 1≤ u_1 < u_2 < ⋯ be the sequence of integers with at most 2 prime factors. Is it true that limsup_k → ∞ u_k+1-u_k/log k=∞?
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Erdős Problem #1142
Are there infinitely many n > 2 such that n - 2^k is prime for all k ≥ 1 with 2^k < n? The only known such n are 4, 7, 15, 21, 45, 75, 105 (OEIS A039669).
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Erdős Problem #1201
Is it true that for every ε,η>0 there exists a k such that the density of n for which P(n(n+1)⋯(n+k))>n^1-ε is at least 1-η (where P(m) is the greatest prime divisor of m)?
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Erdős Problem #1212
Let G be the graph with vertex set those pairs (x,y)∈ ℕ^2 with gcd(x,y)=1, in which we join two vertices if the differ in only one coordinate, and there by ± 1. Is there a path going to infinity on G, say P, such that for all (x,y)∈ P both min(x,y)>1 and at least one of x or y is composite?
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Erdős Problem #141
Let k≥3. Are there k consecutive primes in arithmetic progression?
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Erdős Problem #15
Is it true that Σ_n=1^∞(-1)^nn/p_n converges, where p_n is the sequence of primes? Note: In the problem statement, p_n is the n-th prime, indexed such that p_1=2, p_2=3, …. We 0-index here to reflect how Nat.nth works.
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Erdős Problem #17
Erdős Problem 17. Are there infinitely many cluster primes?
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Erdős Problem #200
Does the longest arithmetic progression of primes in 1,…,N have length o(log N)?
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Erdős Problem #203
Is there an integer m with (m, 6) = 1 such that none of 2^k · 3^ℓ · m + 1 are prime, for any k, ℓ ≥ 0?
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Erdős Problem #233
A conjecture by Heath-Brown: The sum of squares of the first N gaps between consecutive primes behaves like N * (log N)^2.
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Erdős Problem #234
Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c?
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Erdős Problem #236
Let f(n) count the number of solutions to n=p+2^k for prime p and k≥ 0. Show that f(n)=o(log n).
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Erdős Problem #238
Let c₁, c₂ > 0. Is it true that for any sufficiently large x, there exists more than c₁ * log x many consecutive primes ≤ x such that the difference between any two is > c₂?
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Erdős Problem #244
Let C > 1. Does the set of integers of the form p + ⌊ C^k ⌋, for some prime p and k≥ 0, have density >0?
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Erdős Problem #279
Let k≥ 3. Is there a choice of congruence classes a_ppmodp for every prime p such that all sufficiently large integers can be written as a_p+tp for some prime p and integer t≥ k?
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Erdős Problem #375
Is Erdos375Prop true?
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Erdős Problem #428
Is there a set A⊆ ℕ such that, for infinitely many n, all of n-a are prime for all a∈ A with 0 < a < n and liminflvert A∩ [1,x]rvert/π(x)>0?
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Erdős Problem #431
Are there two infinite sets A and B such that A+B agrees with the primes up to finitely many exceptions?
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Erdős Problem #454
Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?
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Erdős Problem #458
Let lcm(1, …, n) denote the least common multiple of 1, …, n. Let p_k be the k-th prime. Is it true that for all k ≥ 1, lcm(1, …, p_k+1-1) < p_k · lcm(1, …, p_k)?
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Erdős Problem #462
Let p(n) denote the least prime factor of n. Is there a constant C>0 such that Σ_x≤ n≤ x+C√(x)(log x)^2p(n)/n≫ 1 for all sufficiently large x?
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Erdős Problem #463
Is there a function f with f(n)→∞ as n→∞ such that, for all large n, there is a composite number m such that n + f(n) < m < n + p(m) Here p(m) is the least prime factor of m.
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Erdős Problem #5
Let C≥ 0. Is there an infinite sequence of n_i such that lim_i→ inftyp_n_i+1-p_n_i/log n_i=C? We formalise "an infinite sequence of n_i" as a strictly monotone sequence of indices n : ℕ → ℕ.
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Erdős Problem #680
Is it true that, for all sufficiently large n, there exists some k such that p(n+k)>k^2+1, where p(m) denotes the least prime factor of m?
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Erdős Problem #681
Erdős problem 681. Is it true that for all large n there exists k such that n + k is composite and p(n+k) > k^2, where p(m) is the least prime factor of m ?
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Erdős Problem #683
Let P(n, k) be the largest prime factor of C(n, k). There exists c > 0 such that P(n, k) ≥ min(n - k + 1, k^1 + c) for all 0 < k ≤ n/2. Erdős stated this for 1 ≤ k ≤ n with the bound min(n-k+1, k^1+c) [Er79d].
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Erdős Problem #779
Erdős Problem #779
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Erdős Problem #850
Can there exist two distinct integers x and y such that x,y have the same prime factors, x+1,y+1 have the same prime factors, and x+2,y+2 also have the same prime factors?
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Erdős Problem #853
Let d_n = p_n+1 - p_n, where p_n is the nth prime. Let r(x) be the smallest even integer t such that d_n = t has no solutions for n ≤ x. Is it true that r(x) → ∞?
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Erdős Problem #855
Erdős Problem 855 (Segal's conjecture): π(x + y) ≤ π(x) + π(y) for all sufficiently large x, y, i.e. for all x, y ≥ N for some N.
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Erdős Problem #890
If ω_k(n) counts the number of distinct prime factors of n which are >k, then is it true that, for every k≥ 1, liminf_n→ ∞Σ_0≤ i < kω_k(n+i)≤ k?
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Erdős Problem #9
Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?
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Erdős Problem #950
Is it true that liminf f(n)=1?
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