Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 7

Problem 855

claims/: The 1 claim page of Problem 855, one per claimant's result; the problem's standing derives from them.

Statement. If $\pi(x)$ counts the number of primes in $[1,x]$ then is it true that (for large $x$ and $y$ )

\pi(x+y) \leq \pi(x)+\pi(y)?

Status. Open.

Source. erdosproblems.com/855, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #855, https://www.erdosproblems.com/855.

References.

[ClJa01] Clark, David A. and Jarvis, Norman C., Dense admissible sequences. Math. Comp. (2001), 1713-1718.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[Er85c] Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp. Section A9 "Patterns of primes", printed p. 40: the prime-pattern conjecture is "incompatible with the well-known conjecture (also due to Hardy & Littlewood)" that $\pi(x+y)\le\pi(x)+\pi(y)$ for all integers $x,y\ge2$ , a display Guy sets between inverted and upright question marks because it "is very likely to be false", with the Montgomery--Vaughan bound. Library home: guy_2004_unsolved_problems_number_theory.
[HeRi73] Hensley, Douglas and Richards, Ian, On the incompatibility of two conjectures concerning primes. (1973), 123-127.
[MoVa73] Montgomery, H. L. and Vaughan, R. C., The large sieve. Mathematika (1973), 119-134.

Formalization. Statement in formal-conjectures.

Current assessment

The standing above derives from the claim page below, and the site's label is also OPEN. The notes below are not independently reviewed. This page records no current literature search or independent assessment of proof coverage.

Hensley and Richards proved in Acta Arithmetica that the prime $k$ -tuples conjecture is incompatible with the inequality: under that conjecture, for every large $x$ there are infinitely many $y$ with $\pi(x+y)>\pi(x)+\pi(y)$ . The result is recorded on their conditional claim page; it settles nothing unconditionally, since the $k$ -tuples conjecture is unproved.

No claim page records the three manuscripts of the OpenAI mathematics release that the library links here, the zero-free half-planes $\operatorname{Re}s>7/8$ and $\operatorname{Re}s>11/12$ for every Dirichlet $L$ -function (card, card) and the uniform exclusion of Landau--Siegel zeros (card): none of them names this problem or claims anything about $\pi(x+y)\le\pi(x)+\pi(y)$ . Their only bearing is on the hypothesis of Granville's conditional interval constructions, recorded on his card, which assume infinitely many Siegel zeros; if the release's claims stand, those constructions have a false hypothesis, and the inequality itself is untouched either way. The claims are unverified here.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (55)

Problem 1210 Hensley and Richards's conditional failure of the inequality problems/primes/E0855/claims Primes additive_combinatorics/erdos_1957_unsolved_problems Problem 1 (p. 291): is pi(x+y) at most pi(x) + pi(y)?Sieving intervals and Siegel zeros Corollary 3 (p. 5): with infinitely many Siegel zeros, admissible sets of length y can have ~ 2y/log y elements A construction of A. Schinzel — many numbers in a short interval without small prime factors Corollary 1 (PDF p. 32): under the prime k-tuple conjecture, pi(x+y) - pi(x) - pi(y) >> x (log x)^{-2} log log log x for some y number_theory/guy_2004_unsolved_problems_number_theory Primes Alkan: A Generalization of the Hardy-Littlewood Conjecture Axler: Some Results on a Conjecture of Hardy and Littlewood Proposition 2.4 (p. 2): pi(m+n) <= pi(m)+pi(n) for all m, n >= 2 with m+n <= 39,708,229,123 Proposition 5.1 (p. 5): explicit expansions of pi(x) give pi(x+y) <= pi(x)+pi(y) for max{5393, cx/log^k x} <= y <= x Theorem 1.1 (p. 2): pi(m+n) <= pi(m)+pi(n) whenever m/1950 <= n <= m Theorem 1.2 (p. 2): for fixed epsilon, pi(m+n) <= pi(m)+pi(n) on epsilon m <= n <= m once m >= exp sqrt(0.3426/log(1+epsilon))Theorem 1.3 (p. 2): pi(m+n) <= pi(m)+pi(n) whenever m >= n >= 2 and n >= c_0 m/log^2 m Theorem 1.4 (p. 2): pi(m+n) <= pi(m)+pi(n) for m+n <= 10^20 and n >= 2 sqrt(m)(1 - 2c_1/(log m + c_1))Theorem 1.5 (p. 2): under RH, pi(m+n) <= pi(m)+pi(n) whenever n >= sqrt(m) log m log(m log^8 m)/(4 pi)Chahal et al.: On the Second Hardy-Littlewood Conjecture Corollary 1.2 and Remark 1.3 (pp. 3-4): under RH, pi(x+y) <= pi(x)+pi(y) for x >= x_eps and (2+eps) x^{1/2} log^2 x/(8 pi) <= y <= x Corollary 1.4 (p. 4): the explicit range of Remark 1.3 under RH verified only up to height T_0 Corollary 1.5 (p. 4): the pairs 2 <= y <= x <= X with pi(x+y) > pi(x)+pi(y) number << X R(2X) log^2 X / log log X Theorem 1.1 (p. 3): pi(x+y) <= pi(x)+pi(y) for x >= x_0 and 3CR(2x) log^2 x / log log x <= y <= x Clark–Jarvis: Dense admissible sequences Conjecture B (p. 1713): π(x+y) − π(y) ≤ π(x), with the functions ϱ and ϱ*Result (p. 1716): ϱ*(x) < π(x) for 2 ≤ x ≤ 1120 and ϱ*(x) ≤ π(x) for 1121 ≤ x ≤ 1426 Result (p. 1717): ϱ*(5380) ≥ 715 > π(5380) and ϱ*(4916) ≥ 657 > π(4916)Table 5 (p. 1718): admissible sequences with S(x) > 2π(x/2), against Erdős's Conjecture C Dusart: Sur la conjecture π(x+y)≤π(x)+π(y)primes/hensley_1974_primes_intervals Theorem: ρ*(x) − π(x) → ∞, and ρ*(x) − π(x) ≥ (log 2 − ε) x/(log x)² for large x Johnston–Yang: Some explicit estimates for the error term in the prime number theorem Corollary 1.2 (p. 2): the Table 1 bounds for theta(x) - x, with A_1 = A + 0.01 Corollary 1.3 (p. 2): |pi(x) - li(x)| <= 9.59x(log x)^0.515 exp(-0.8274 sqrt(log x)) for x >= 2 Lemma 2.2 (p. 4): square-root error bounds for psi, theta and pi up to 2.169 x 10^25 Theorem 1.1 (p. 2): explicit de la Vallée Poussin-shaped bounds for psi(x) - x Theorem 1.4 (p. 2): Vinogradov–Korobov-shaped explicit bounds for psi, theta and pi, x >= 23 primes/openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12 Theorem 1.1: zero-free half-plane Re s > 11/12 for finite-order Hecke L-functions over Q(√−3), hence for Dirichlet L-functions and zeta primes/openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_7_8 Theorem 1.1: no zero of any finite-order Hecke L-function over Q(sqrt(-3)) or any Dirichlet L-function in Re s > 7/8 primes/openai_2026_uniform_exclusion_landau_siegel_zeros Theorem 1: (1 − β) log q ≥ c for every real zero of a real L(s, χ)Richards: On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem Corollary 1.10 (p. 425): under the prime k-tuples conjecture, every large x has infinitely many y with π(y+x) − π(y) > π(x)Definition 1.7 (p. 424): ρ*(x), the largest admissible tuple in an interval of length x Proposition 1.9 and Corollary (p. 424): under the prime k-tuples conjecture, ρ*(x) is the largest prime count of infinitely many intervals of length x

Graph