Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|x...Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.展开更多
For any positive integer n, the famous Smarandache power function SP（n） is defined as the smallest positive integer m such that n｜m^m, where m and n have the same prime divisors. The main purpose of this paper is u...For any positive integer n, the famous Smarandache power function SP（n） is defined as the smallest positive integer m such that n｜m^m, where m and n have the same prime divisors. The main purpose of this paper is using the elementary methods to study the positive integer solutions of an equation involving the Smarandache power function SP（n） and obtain some interesting results. At the same time, we give an open problem about the related equation.展开更多
The Diophantine equation X（ X ＋ 1 ） （ X ＋ 2 ） （ X ＋ 3 ） = 14Y（ Y ＋ 1 ） （ Y ＋ 2 ） （ Y ＋ 3 ） still remains open. Using recurrence sequence, Maple software, Pell equation and quadraric residue, this pap...The Diophantine equation X（ X ＋ 1 ） （ X ＋ 2 ） （ X ＋ 3 ） = 14Y（ Y ＋ 1 ） （ Y ＋ 2 ） （ Y ＋ 3 ） still remains open. Using recurrence sequence, Maple software, Pell equation and quadraric residue, this paper proved it has only two positive integer solutions, i. e., （X,Y） = （5,2） ,（7,3）.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
文摘Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.
基金Supported by the Natural Science Foundation of China（10671155）
文摘For any positive integer n, the famous Smarandache power function SP（n） is defined as the smallest positive integer m such that n｜m^m, where m and n have the same prime divisors. The main purpose of this paper is using the elementary methods to study the positive integer solutions of an equation involving the Smarandache power function SP（n） and obtain some interesting results. At the same time, we give an open problem about the related equation.
基金The Natural Science Foundation of Chongqing University of Post and Telecommunications (No.A2008-40)
文摘The Diophantine equation X（ X ＋ 1 ） （ X ＋ 2 ） （ X ＋ 3 ） = 14Y（ Y ＋ 1 ） （ Y ＋ 2 ） （ Y ＋ 3 ） still remains open. Using recurrence sequence, Maple software, Pell equation and quadraric residue, this paper proved it has only two positive integer solutions, i. e., （X,Y） = （5,2） ,（7,3）.
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.
