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TOT 376 1993 Spring A S4 a_{n+1} \ne sum k_n *a_n

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June 10, 2024
number theorycombinatorics

Problem Statement

Positive integers are written on the blackboard one after another. The next integer an+1a_{n+1} (to be written after a1a_1,a2a_2,......,ana_n) is an arbitrary integer not representable as a sum of several previous integers taken one or more times (i.e. an+1a_{n+1} is not of the form k1∗ai+k2∗a2+...+kn∗ank_1 *a_i + k_2 *a_2 + ... + k_n *a_n wherek1 k_1, k2k_2,...., knk_n are non-negative integers). Prove that the process of writing cannot be infinite.
(A Belov)
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