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numbers not the sum of 2+ terms in an arithmetic sequence

Source: Serbia TST 1999 P4

May 14, 2021
number theoryarithmetic sequence

Problem Statement

For a natural number dd, MdM_d denotes the set of natural numbers which are not representable as the sum of at least two consecutive terms of an arithmetic progression with the common difference d whose terms are integers. Prove that each c∈M3c\in M_3 can be written in the form c=abc=ab, where a∈M1a\in M_1 and b∈M2∖{2}b\in M_2\setminus\{2\}.
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