Combinatorics and Graph Theory

    

Lonely Runner Conjecture Proof

Authors: Bhaskar Kumar

The Lonely Runner Conjecture, independently proposed by Wills (1967) and Cusick (1973), asserts that for any n runners on a unit circular track moving at distinct integer speeds, there exists a time at which each runner is at distance at least 1/n from a fixed reference runner. We provide a complete proofby reformulating the problem in terms of simultaneous residue conditions and proving existence via an inductive construction using the least common multiple and arithmetic properties of congruences. The proof is elementary, relying only on the Chinese Remainder Theorem and basic number-theoretic tools.

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[v1] 2026-03-17 00:22:13

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