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Infinitely numbers with property P - ILL 1990 FRA2

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September 18, 2010

number theoryAdditive Number TheoryAdditive combinatoricsIMO ShortlistIMO Longlist

Problem Statement

We call an integer

k \geq 1

having property

P

, if there exists at least one integer

m \geq 1

which cannot be expressed in the form

m = \varepsilon_1 z_1^k + \varepsilon_2 z_2^k + \cdots + \varepsilon_{2k} z_{2k}^k

, where

z_i

are nonnegative integer and

k

or

-1

,

i = 1, 2, \ldots, 2k

. Prove that there are infinitely many integers

k

having the property

P.

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