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Updated Oct 8

Problem 1200

Statement. There exists a constant $C$ such that for all large $x$ there is a collection of primes $p_1<\ldots<p_k<x$ with $\sum\frac{1}{p_i}<C$ together with a system of congruences $a_i\pmod{p_i}$ such that every integer $n<x$ satisfies at least one of these congruences.

Status. Open.

Source. erdosproblems.com/1200, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1200, https://www.erdosproblems.com/1200.

References.

[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[ErRu80] Erdős, P. and Ruzsa, I. Z., On the small sieve. I. Sifting by primes. J. Number Theory (1980), 385-394.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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 (10)

Primes number_theory/erdos_1980_survey_problems_combinatorial_number_theory Primes primes/erdos_1980_small_sieve Problem 2 (p. 386): does sifting by arbitrary residue classes of primes of reciprocal sum at most K leave at least c(K) x integers?primes/ruzsa_1982_small_sieve_ii_sifting_composite_numbers Theorem III (p. 262): 1/2 < μ(x) < log(5/2) + o(1/x) for covering systems with distinct moduli in (1, x]primes/warlimont_1991_problem_posed_i_z_ruzsa Equation (1) (p. 54): the linear-programming relaxation nu*(n) of Ruzsa's covering cost equals log(2^5 3^6/23^3) + O(1/n)Inequality (2) (p. 54): the 0-1 relaxation nu(n) exceeds its linear-programming relaxation nu*(n) by O(1/n)

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