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Updated Oct 8

Problem 25

claims/: The 2 claim pages of Problem 25, one per claimant's result; the problem's standing derives from them.

Statement. Let $1\leq n_1<n_2<\cdots$ be an arbitrary sequence of integers, each with an associated residue class $a_i\pmod{n_i}$ . Let $A$ be the set of integers $n$ such that for every $i$ either $n<n_i$ or $n\not\equiv a_i\pmod{n_i}$. Must the logarithmic density of $A$ exist?

Status. Open.

Source. erdosproblems.com/25, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #25, https://www.erdosproblems.com/25.

Formalization. Statement in formal-conjectures.

Current assessment

The question, in the site's formulation accessed, asks whether the set $A$ of integers that avoid, for every modulus $n_i$ not exceeding them, the residue class $a_i$ modulo $n_i$ must have a logarithmic density; the site labels the problem OPEN and notes it as a special case of Problem 486. The standing is open with no full claim. One pending partial claim is recorded: Chojecki's note of 19 March 2026 ([[problems/integer_sequences/E0025/claims/2026_03_19_chojecki|claim page]]) proves that $A$ has natural, hence logarithmic, density when $\sum_i1/n_i<\infty$ and when the $n_i$ are pairwise coprime, and the same note reduces the general case to an unproved uniform harmonic estimate for its quotient sieves ([[problems/integer_sequences/E0025/claims/2026_03_19_chojecki_conditional|conditional page]]). The note is unrefereed and neither page counts toward the standing. The zero-residue case, every $a_i=0$ , makes $A$ the complement of the set of multiples of the $n_i$ , whose logarithmic density exists by Theorem 1 of Davenport and Erdős (card); that deduction is a remark made here, which the paper does not state for this problem and the site does not credit, so it has no claim page. Wang's proposed counterexample to Problem 486, a sieve with several forbidden residues per modulus and no logarithmic density, does not transfer to a single residue per modulus, as its card explains (card). No literature search beyond these sources and no independent review of any proof is recorded.

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 (42)

An elementary proof from Davenport-Erdős and Rogers Chojecki's two unconditional special cases Chojecki's conditional reduction to a quotient-sieve estimate problems/integer_sequences/E0025/claims Sequences and Densities of Integers Covering systems and density Sarnak’s Program for Erdős Sieves. Part II: Measure Systems and Applications library/covering_systems/balister_2018_erdos_covering_problem_density_uncovered_set On the density of certain sequences of integers library/integer_sequences/chojecki_2026_truncated_congruence_sieves_erdos_problem_25 On sequences of positive integers On sequences of positive integers library/integer_sequences/filaseta_2007_sieving_large_integers_covering_systems_congruences library/covering_systems/hough_2015_solution_minimum_modulus_problem_covering_systems Wang: A Proposed Solution to Erdős Problem 486 covering_systems/balister_2018_erdos_covering_problem_density_uncovered_set covering_systems/hough_2015_solution_minimum_modulus_problem_covering_systems divisors/davenport_1951_sequences_positive_integers Lower and logarithmic density of a set of multiples Sequences and Densities of Integers Sarnak’s Program for Erdős Sieves. Part II: Measure Systems and Applications On the density of certain sequences of integers The §7 construction (p. 340): a primitive set whose multiples have no natural density Theorem 1 (p. 339): the dyadic divisor-window densities have average zero integer_sequences/chojecki_2026_truncated_congruence_sieves_erdos_problem_25 Conjecture 5.1 (p. 6): a uniform harmonic estimate for the quotient sieves Corollary 5.3 (p. 7): the entropy terms of the first-kill sets total O(log log X)Lemma 2.1 (p. 3): finite truncations of the sieve are eventually periodic Proposition 4.1 (pp. 4--5): the first-kill sets partition the sifted-out integers Proposition 4.2 (pp. 5--6): each first-kill set is a dilate of a periodic quotient sieve Proposition 6.1 (p. 10): no vanishing bound for the logarithmic mass of the tail union Proposition 6.3 (pp. 10--11): a prime-power tower whose top first-kill set has a large early harmonic spike Theorem 3.1 (pp. 3--4): the sieve has natural density when the reciprocal moduli are summable Theorem 3.2 (p. 4): the sieve has natural density when the moduli are pairwise coprime Theorem 5.4 (pp. 7--9): conditional reduction of Problem 25 to the quotient-sieve estimate integer_sequences/davenport_1936_sequences_positive_integers integer_sequences/filaseta_2007_sieving_large_integers_covering_systems_congruences Lemma 2.1 (p. 6): uncovered density is at least alpha minus beta Lemma 3.4 (p. 14): the smooth-rough lower bound for the uncovered density Theorem 2 (p. 15): positive uncovered density for large moduli of bounded multiplicity Theorem 6 (p. 22): in long windows the uncovered density can fall below 1/K Wang: A Proposed Solution to Erdős Problem 486

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