MathDB
A representation

Source: MEMO 2018 T4

September 2, 2018
number theoryalgebra

Problem Statement

Let nn be a positive integer and u1,u2,⋯ ,unu_1,u_2,\cdots ,u_n be positive integers not larger than 2k,2^k, for some integer k≥3.k\geq 3. A representation of a non-negative integer tt is a sequence of non-negative integers a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n such that t=a1u1+a2u2+⋯+anun.t=a_1u_1+a_2u_2+\cdots +a_nu_n. Prove that if a non-negative integer tt has a representation,then it also has a representation where less than 2k2k of numbers a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n are non-zero.
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