The Number of Solutions for a Class of Exponential Diophantine Equations

Abstract

Abstract: Let a, b, c, k be posi tive integers such taht a+ b= ck , gcd( a, b ) = 1, c ∈ { 1, 2, 4} , k> 1 and k is odd if c= 1 or 2. Further let ε= ( a+ - b ) / c and ε= ( a- - b ) / c. In this paper we prove that if ( a, b , c,k )≠( 1, 7, 4, 2 ) or ( 3, 5, 4, 2 ) , then there exist at mos t one positive odd integer n satisfying( ε ⁿ- εⁿ) / ( ε- ε) = 1 with n> 1.Moreover , such n must be an odd prime satisfying n< 1+ ( 2log π) / log k+ 2563. 43( 1+ 21. 96π/ log k) .

Key words: exponential diopnantine equation, number of solutions, upper bound

LE Mao-Hua. The Number of Solutions for a Class of Exponential Diophantine Equations[J]. journal6, 2000, 21(3): 27-32.

References

[ 1] AP RY R. Sur Une q uation Diophantienne[ J] . C R Acad Sci. Pari s S r A, 1960, 251: 1266~ 1264

[ 2] AP RY R. Sur Une q uation Diophantienne[ J] . C R Acad Sci. Pari s S r A, 1960, 251: 1451~ 1452

[ 3] BENDER E, HERZBERG N. Some Diophantine Equations Related to the Quadratic Form ax2+ by2[ A] . In: Studies in Algebra andNumber Theory ( Ed. Rota G- C) [ C] . San Diego: Academic Press, 1979. 219~ 272

[ 4] BEUKERS F. On the Generalized Ramanujan- Nagell Equation [ J] . Acta. Arith. , 1980/ 1981, 38: 389~ 410

[ 5] COHN J H E. The Diophantine Equation x2+ c = yn[ J] . Acta. Ari th. , 1993, 65: 367~ 381

[ 6] LE M- H. On the Divisibility of the Class Number of the Imaginary Quadratic Fidle Q( a2- 4km) [ J] . Acta. Math. Sinica( NS) ,1989, 5( 1) : 80~ 86

[ 7] LEM H. Sur Le Nombre de Solutions de I quation Di ophantienne x2+ D= pn[ J] . C R Acad. Sci. Paris S r I Math. , 1993, 317:135~ 138

[ 8] LE M H. On the Diophantine Equations D1 x2+ D2= 2n+ 2[ J] .Acta. Arith. , 1996, 64( 1) : 29~ 41

[ 9] LE M H. On the Diophantine Equations d1x2+ 22md2= ynand d1x2+ d2= 4yn[ J] . Proc. Amer. Math. Soc. , 1993, 118( 1) : 67~70

[ 10] LE M H. A Note on the Generalized Ramanujan- Nagell Equation[ J] . J. Number Theory, 1995, 50( 2) : 193~ 201

[ 11] LE M H. A Note on the Number of Solutions of the Generaized Ramanujan- Nagell Equation D1x2+ D2= 4pn[ J] . J. NumberThe -ory, 1997, 62( 1) : 100~ 106

[ 12] LE M H. The Di ophantine Equation x2+ 2m= yn[ J] . Chinese Sci. Bull. , 1997. 42

[ 13] LE M H. A Note on the Diophantine Equati on D1x2+ D2= 2yn[ J] . Publ. Math. Debrecen, 1997, 51( 1~ 2) : 191~ 198

[ 14] LE M H. On the Diophantine Equation ( x3- 1) / ( x- 1)= ( yn- 1) / ( y - 1) [ J] . Trans. Amer. Math. Soc. , 1999, 351( 3) : 1063~ 1074

[ 15] NAGELL T. the Diophantine Equation x2+ 7= 2n[ J] . Ark. Math. , 1960, 4: 185~ 187

[ 16] LIDL R, NIEDERREITER H. Finite Fields[ J] . MA: Addison- Wesley, Reading, 1983.

[ 17] MORDELL L J. Di ophantine Equations[M] . London: Academic Press, 1969.

[ 18] VOUTIER P M. Primitive Divisors of Lucas and Lehmer Sequences[ J] . Math. Comp. , 1995, 64: 869~ 888

[ 19] LAURENT M, MIGNOTTE M, NES TERENKO Y. Formes Li naires en Deux Logarithmes et Dterminants D interpolation[ J] . J.Number Theory, 1995, 55: 285~ 321

[ 20] 华罗庚数论导引[M] 北京: 科学出版社, 1979

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