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On a Problem of Brocard
Author(s) -
Gica Alexandru,
Panaitopol Laurenţiu
Publication year - 2005
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/s0024609305004595
Subject(s) - mathematics , conjecture , degree (music) , integer (computer science) , mathematics subject classification , combinatorics , polynomial , finitely generated abelian group , discrete mathematics , mathematical analysis , physics , computer science , acoustics , programming language
It is proved that, if P is a polynomial with integer coefficients, having degree 2, and 1 > ε > 0, then n ( n − 1) … ( n − k + 1) = P ( m ) has only finitely many natural solutions ( m,n,k ), n ⩾ k > n ε, provided that the abc conjecture is assumed to hold under Szpiro's formulation. 2000 Mathematics Subject Classification 11D75, 11J25, 11N13.