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
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).
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）.
