In this paper, we give out the formula of number of primes no more than any given n (n ∈ Z<sup>+</sup>, n > 2). At the same time, we also show the principle, derivation process of the formula and appli...In this paper, we give out the formula of number of primes no more than any given n (n ∈ Z<sup>+</sup>, n > 2). At the same time, we also show the principle, derivation process of the formula and application examples, it is usually marked with π(n), which is: that is: where “[ ]” denotes taking integer. r = 1,2,3,4,5,6;s<sub>x</sub> = s<sub>1</sub>,s<sub>2</sub>,...,s<sub>j</sub>,s<sub>h</sub>;s1</sub>,s2</sub>,...,s<sub>j</sub>,,s<sub>h </sub><sub>= 0,1,2,3,....</sub>As i ≥ 2, 2 ≤ s<sub>x </sub>≤ i-1 (x=1,2,...,j,h).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
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
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
Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 ＋ b^2 = c^3 and b is an odd prime, then the equation a^x ＋ b^y = c^z has only the positive integer solution （x, y, z） = （2,2,3）.
文摘In this paper, we give out the formula of number of primes no more than any given n (n ∈ Z<sup>+</sup>, n > 2). At the same time, we also show the principle, derivation process of the formula and application examples, it is usually marked with π(n), which is: that is: where “[ ]” denotes taking integer. r = 1,2,3,4,5,6;s<sub>x</sub> = s<sub>1</sub>,s<sub>2</sub>,...,s<sub>j</sub>,s<sub>h</sub>;s1</sub>,s2</sub>,...,s<sub>j</sub>,,s<sub>h </sub><sub>= 0,1,2,3,....</sub>As i ≥ 2, 2 ≤ s<sub>x </sub>≤ i-1 (x=1,2,...,j,h).
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.
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.
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the National Natural Science Foundation of China (No.10271104)the Guangdong Provincial Natural Science Foundation (No.04011425)
文摘Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 ＋ b^2 = c^3 and b is an odd prime, then the equation a^x ＋ b^y = c^z has only the positive integer solution （x, y, z） = （2,2,3）.
