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On a question of Sárkozy and Sós for bilinear forms
Author(s) -
Cilleruelo Javier,
Rué Juanjo
Publication year - 2009
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn123
Subject(s) - mathematics , sequence (biology) , constant (computer programming) , bilinear interpolation , dirac (video compression format) , function (biology) , representation (politics) , pure mathematics , discrete mathematics , statistics , quantum mechanics , computer science , law , genetics , physics , evolutionary biology , politics , political science , neutrino , biology , programming language
We prove that, if 2 ⩽ k 1 ⩽ k 2 , then there is no infinite sequence of positive integers such that the representation function r ( n ) = #{( a , a ′): n = k 1 a + k 2 a ′, a , a ′ ∈ } is constant for n large enough. This result completes the previous work of Dirac and Moser for the special case k 1 = 1 and answers a question posed by Sárkozy and Sós.