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Problem 275
claims/: The 2 claim pages of Problem 275, one per claimant's result; the problem's standing derives from them.
Statement. If a finite system of congruences ${ a_i\pmod{n_i} : 1\leq i\leq r}$ (the are not necessarily distinct) covers consecutive integers then it covers all integers.
Status. PROVED (LEAN). The proof of Crittenden and Vanden Eynden and the short proof of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, which the Lean proof follows, are recorded on the first and the second claim page.
Source. erdosproblems.com/275, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #275, https://www.erdosproblems.com/275.
References.
- [BBMST20b] Balister, P. and Bollobás, B. and Morris, R. and Sahasrabudhe, J. and Tiba, M., Covering intervals with arithmetic progressions. Acta Math. Hungar. (2020), 197-200.
- [CrVE70] Crittenden, R. B. and Vanden Eynden, C. L., Any arithmetic progressions covering the first integers cover all integers. Proc. Amer. Math. Soc. (1970), 475-481.
Formalization. Statement in formal-conjectures.
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.
- klein_2023_jth_smallest_modulus_covering_system
- klein_2023_jth_smallest_modulus_covering_system / claim_2_1
- simpson_1997_crittenden_vanden_eynden_coverings
- simpson_1997_crittenden_vanden_eynden_coverings / theorem_14
- sun_1995_covering_integers_arithmetic_sequences
- sun_1996_covering_integers_arithmetic_sequences_ii
- sun_1996_covering_integers_arithmetic_sequences_ii / theorem_1