In this paper,we construct new examples of hyperbolic metasurfaces in CP^(3) and CP^(4),and discusses the existence of solutions for a class of Fermat type functional equations.
The main purpose of this paper is to try to find all entire solutions of the Fermat type difference-differential equation[p1(z)f(z+c)]^(2)+[p2(z)f(z)+p3(z)f′(z)]^(2)=p(z);or[p1(z)f(z)]^(2)+[p2(z)f′(z)+p3(z)f(z+c)]^(...The main purpose of this paper is to try to find all entire solutions of the Fermat type difference-differential equation[p1(z)f(z+c)]^(2)+[p2(z)f(z)+p3(z)f′(z)]^(2)=p(z);or[p1(z)f(z)]^(2)+[p2(z)f′(z)+p3(z)f(z+c)]^(2)=p(z)or[p1(z)f′(z)]^(2)+[p2(z)f(z+c)+p3(z)f(z)]^(2)=p(z);where c is a nonzero complex number,p1;p2 and p3 are polynomials in C satisfying p1p3■0;and p is a nonzero irreducible polynomial in C.展开更多
Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A<sup>n</sup> + B<sup>n</sup> = C<sup>n</sup>, in which n is a natural number great...Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A<sup>n</sup> + B<sup>n</sup> = C<sup>n</sup>, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A<sup>2</sup> + B<sup>2</sup> + C<sup>2</sup> + D<sup>2</sup> + so on =A<sub>n</sub><sup>2 </sup>where all are natural numbers.展开更多
本文得到了Fermat方程解之间的一些最好的不等式,例如Fermat方程x^n+y^n=z^n,O＜x＜y＜z,n＞2 有整数解时推出①对 Vm∈[2,n-1]均有x^m＞nz^(m-1)+(n-m)z^(m-2)+[(n-m)(n+log2)]/2 z^(m-3)-sum fromi=i to m (m/i) z^(m-1)(-1)~1;②x^n...本文得到了Fermat方程解之间的一些最好的不等式,例如Fermat方程x^n+y^n=z^n,O＜x＜y＜z,n＞2 有整数解时推出①对 Vm∈[2,n-1]均有x^m＞nz^(m-1)+(n-m)z^(m-2)+[(n-m)(n+log2)]/2 z^(m-3)-sum fromi=i to m (m/i) z^(m-1)(-1)~1;②x^n＞[(n+log2)/2]z^(n-1),等等.同时对若干类型的素数指数,我们还证明了Fermat大定理第一情形成立.展开更多
As expounded in some recent mathematical conferences, this research on that amazing source of algebraic ideas known as Fermat＇s equation is aimed to prove how Fermat triples can be limited until the impossible existe...As expounded in some recent mathematical conferences, this research on that amazing source of algebraic ideas known as Fermat＇s equation is aimed to prove how Fermat triples can be limited until the impossible existence through a criterion of incompatible parities related to unexplored properties of the binomial coefficients. In this paper, the authors use a technique based on the analysis of four numbers and their internal relations with three basic compulsory factors. It leads to the practical impossibility to find any triple of natural numbers candidate to satisfy Fermat＇s equation, because when the authors try to meet a condition between parity and range the authors are compelled to violate the other one, so that they are irreducibly alternative. In particular, there is a parity violation when the authors choose all the basic factors in the allowed range and the authors obtain exceeding values of one of the involved variables when the authors try to restore the parity. Since Fermat＇s last theorem would consequently be demonstrated, many readers could recall the never found elementary proof of FLT （Fermat＇s last theorem） claimed by Pierre de Fermat. The authors are not encouraging such an interpretation because this paper is intended as a journey into Fermat＇s equation and the reader＇s attitude should be towards the algebraic achievements here proposed, with their possible hidden flaws and future developments, rather than to legendary problems like Fermat＇s riddle.展开更多
基金Supported by the National Natural Foundation of China(Grant No.12361028)the Foundation of Education Department of Jiangxi(Grant Nos.GJJ212305 and GJJ2202228)。
文摘In this paper,we construct new examples of hyperbolic metasurfaces in CP^(3) and CP^(4),and discusses the existence of solutions for a class of Fermat type functional equations.
基金Supported by the National Natural Science Foundation of China(11871260,11761050)the Jiangxi Natural Science Foundation(#20232ACB201005)+1 种基金the Shandong Natural Science Foundation(#ZR2024MA024)Doctoral Startup Fund of Jiangxi Science and Technology Normal University(#2021BSQD30).
文摘The main purpose of this paper is to try to find all entire solutions of the Fermat type difference-differential equation[p1(z)f(z+c)]^(2)+[p2(z)f(z)+p3(z)f′(z)]^(2)=p(z);or[p1(z)f(z)]^(2)+[p2(z)f′(z)+p3(z)f(z+c)]^(2)=p(z)or[p1(z)f′(z)]^(2)+[p2(z)f(z+c)+p3(z)f(z)]^(2)=p(z);where c is a nonzero complex number,p1;p2 and p3 are polynomials in C satisfying p1p3■0;and p is a nonzero irreducible polynomial in C.
文摘Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A<sup>n</sup> + B<sup>n</sup> = C<sup>n</sup>, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A<sup>2</sup> + B<sup>2</sup> + C<sup>2</sup> + D<sup>2</sup> + so on =A<sub>n</sub><sup>2 </sup>where all are natural numbers.
文摘本文得到了Fermat方程解之间的一些最好的不等式,例如Fermat方程x^n+y^n=z^n,O＜x＜y＜z,n＞2 有整数解时推出①对 Vm∈[2,n-1]均有x^m＞nz^(m-1)+(n-m)z^(m-2)+[(n-m)(n+log2)]/2 z^(m-3)-sum fromi=i to m (m/i) z^(m-1)(-1)~1;②x^n＞[(n+log2)/2]z^(n-1),等等.同时对若干类型的素数指数,我们还证明了Fermat大定理第一情形成立.
文摘As expounded in some recent mathematical conferences, this research on that amazing source of algebraic ideas known as Fermat＇s equation is aimed to prove how Fermat triples can be limited until the impossible existence through a criterion of incompatible parities related to unexplored properties of the binomial coefficients. In this paper, the authors use a technique based on the analysis of four numbers and their internal relations with three basic compulsory factors. It leads to the practical impossibility to find any triple of natural numbers candidate to satisfy Fermat＇s equation, because when the authors try to meet a condition between parity and range the authors are compelled to violate the other one, so that they are irreducibly alternative. In particular, there is a parity violation when the authors choose all the basic factors in the allowed range and the authors obtain exceeding values of one of the involved variables when the authors try to restore the parity. Since Fermat＇s last theorem would consequently be demonstrated, many readers could recall the never found elementary proof of FLT （Fermat＇s last theorem） claimed by Pierre de Fermat. The authors are not encouraging such an interpretation because this paper is intended as a journey into Fermat＇s equation and the reader＇s attitude should be towards the algebraic achievements here proposed, with their possible hidden flaws and future developments, rather than to legendary problems like Fermat＇s riddle.
基金The work of authors were partially supported by NSFC of Shandong(ZR2018MA014)PCSIRT(IRT1264)The Fundamental Research Funds of Shandong University(2017JC019)。
