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http://dx.doi.org/10.1142/S0218196799000199

has been cited by the following article:

• TITLE: Diophantine Equations and the Freeness of Möbius Groups

  AUTHORS: Marin Gutan

  KEYWORDS: Diophantine Equation, Möbius Groups, Free Group

  JOURNAL NAME: Applied Mathematics, Vol.5 No.10, May 23, 2014

  ABSTRACT: Let p and q be two fixed non zero integers verifying the condition gcd(p,q) = 1. We check solutions in non zero integers a1,b1,a2,b2 and a3 for the following Diophantine equations: (B1) (B2) . The equations (B1) and (B2) were considered by R.C. Lyndon and J.L. Ullman in [1] and A.F. Beardon in [2] in connection with the freeness of the M?bius group generated by two matrices of namely and where .They proved that if one of the equations (B1) or (B2) has solutions in non zero integers then the group is not free. We give algorithms to decide if these equations admit solutions. We obtain an arithmetical criteria on p and q for which (B1) admits solutions. We show that for all p and q the equations (B1) and (B2) have only a finite number of solutions.