A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primit...A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.展开更多
In this paper, we study the quantic Diophantine equation （1） with elementary geometry method, therefore all positive integer solutions of the equation （1） are obtained, and existence of Heron triangle whose median...In this paper, we study the quantic Diophantine equation （1） with elementary geometry method, therefore all positive integer solutions of the equation （1） are obtained, and existence of Heron triangle whose median lengths are all positive integer are discussed here.展开更多
LET Z,N,Q be the sets of integers, positive integers and rational numbers, respectively. The solutions (x, y, m, n) of the exponential Diophantine equation X^2+2~m=y^n,x,y,m,n∈N,2y,n【2 (1)are connected with many que...LET Z,N,Q be the sets of integers, positive integers and rational numbers, respectively. The solutions (x, y, m, n) of the exponential Diophantine equation X^2+2~m=y^n,x,y,m,n∈N,2y,n【2 (1)are connected with many questions in number theory and combinatorial theory. In the recent fifty years, there were many papers concerned with the equation written by Ljunggren, Nagell, Brown, Toyoizumi and Cohn. In 1986, ref. [1] claimed that all solutions of (1) had been determined. However, we have not seen the proof so far. Therefore, the solution of (1) has not been found yet. In this note, using Baker’s method, we prove the following result.展开更多
文摘A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.
基金Foundation item: Supported by the Natural Science Foundation of China(10271104)Supported by the Natural Science Foundation of Education Department of Sichuan Province(2004B25)
文摘In this paper, we study the quantic Diophantine equation （1） with elementary geometry method, therefore all positive integer solutions of the equation （1） are obtained, and existence of Heron triangle whose median lengths are all positive integer are discussed here.
文摘LET Z,N,Q be the sets of integers, positive integers and rational numbers, respectively. The solutions (x, y, m, n) of the exponential Diophantine equation X^2+2~m=y^n,x,y,m,n∈N,2y,n【2 (1)are connected with many questions in number theory and combinatorial theory. In the recent fifty years, there were many papers concerned with the equation written by Ljunggren, Nagell, Brown, Toyoizumi and Cohn. In 1986, ref. [1] claimed that all solutions of (1) had been determined. However, we have not seen the proof so far. Therefore, the solution of (1) has not been found yet. In this note, using Baker’s method, we prove the following result.
基金Supported by the NSF of China(11126173)Anhui Province Natural Science Foundation(1208085QA02)+1 种基金the NSF of China(10901002)the NSF of Anhui Province Education Committee(KJ2011Z151)
基金Supported by the National Natural Science Foundation of China(10271104)the Guangdong Provincial Natural Science Foundation(011781)the Natural Science Foundation of Education Department of Guangdong Province(0161)
