Applications Of The Theorem About The Existence Of Primitive Divisors Of Lucas And Lehmer Numbers To Exponential Diophantine Equations

This thesis is to study the integer solutions and the number of theinteger solutions of some exponential diophantine equations by applyingthe deep theorem of Bilu, Hanrot and Voutier about the existence ofprimitive divisors of Lucas and Lehmer numbers, some fine results on therepresentation of the solutions of quadratic Diophantine equations and theclass number of quadratic field.In Chapter 1,we study some generalized Ramanujan-Nagell equatio-ns and have obtained the following results.1. Let D＞2 be an integer distinct from a power of 2 and let p bean odd prime not dividing D, we prove that only if D=3, p=13, then thediophantine equation x²+D^m=pⁿ has exactly two solutions (x,m,n)with 2|m. Otherwise, the equation has at most one positive integersolution with 2|m and this result corrects an inaccuracy of Bugeaud.2. Let D＞2 be an integer distinct from a power of 2 and let p bean odd prime not dividing D, we prove that the diophantine equationx²+D^m=pⁿ has at most two positive integer solutions (x,m,n) and thisresult improves some results obtained by Bugeaud and Le Maohua.3. We prove that the diophantine equation x²+(3a²Â±1)^m=(4a²Â±1)ⁿhas exactly two positive integer solutions (x,m,n) when 3a²Â±1 is anodd prime or an odd prime power. In Chapter 2, we study the diophantine equation A^x+B^y=C^z andhave obtained the following results.1. Let A=|m(m⁴-10m²+5)|,B=5m⁴-10m²+1,C=m²+1,2|m＞0, weprove that the only positive integer solution of the diophantine equationA^x+B^y=C^z is (x,y,z)=(2,2,5), this result improves some resultsobtained by Terai, Cao Zhenfu, Dong Xiaolei and Le Maohua.2. Let a≥2 be a positive integer, we prove that the only positiveinteger solution of the diophantine equation a^2x+(3a²Â±1)^y=(4a²Â±1)^z is(x,y,z)=(1,1,1)。3. Let a≥2 be an integer, we prove that the only positive integersolution of the diophantine equation (8a³+3a)^2x+(3a²Â±1)^y=(4a²Â±1)^z is(x,y,z)=(1,1,3)。In Chapter 3, we study the diophantine equation ax²+D^m=pⁿ andhave generalized some results obtained in Chapter 1.Let a＞1 and let p be an odd prime not dividing D, we provethat if a is not a square, then1. the exponential diophantine equation ax²+D^m=pⁿ has at mosttwo positive integer solutions (x,m,n) except some special cases.2. the exponential Diophantine equation ax²+D^m=pⁿ has at mostthree positive integer solutions (x,m,n).