Define the total number of distinct prime factors of an odd perfect number n asω(n). We prove that if n is an odd perfect number which is relatively prime to 3 and 5 and7, then ω(n) ≥ 107. And using this result, we...Define the total number of distinct prime factors of an odd perfect number n asω(n). We prove that if n is an odd perfect number which is relatively prime to 3 and 5 and7, then ω(n) ≥ 107. And using this result, we give a conclusion that the third largest prime factor of such an odd perfect number exceeds 1283.展开更多
The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact...The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.展开更多
Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly...Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.展开更多
设D=multiply from i=1 to s p_i(s≥2),p_i=1(mod 6)(1≤i≤s)为不同的奇素数.关于Diophantine方程x^3-1=Dy^2的初等解法至今仍未解决.主要利用同余式、平方剩余、Pell方程的解的性质、递归序列,证明了q≡7(mod 24)为奇素数.(q/13)=-1...设D=multiply from i=1 to s p_i(s≥2),p_i=1(mod 6)(1≤i≤s)为不同的奇素数.关于Diophantine方程x^3-1=Dy^2的初等解法至今仍未解决.主要利用同余式、平方剩余、Pell方程的解的性质、递归序列,证明了q≡7(mod 24)为奇素数.(q/13)=-1时,Diophantine方程x^3-1=13qy^2仅有整数解(x,y)=(1,0).展开更多
基金Foundation item： Supported by the Science Foundation of Kashgar Teacher＇s College（112390）
文摘Define the total number of distinct prime factors of an odd perfect number n asω(n). We prove that if n is an odd perfect number which is relatively prime to 3 and 5 and7, then ω(n) ≥ 107. And using this result, we give a conclusion that the third largest prime factor of such an odd perfect number exceeds 1283.
文摘The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.
文摘Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.
文摘设D=multiply from i=1 to s p_i(s≥2),p_i=1(mod 6)(1≤i≤s)为不同的奇素数.关于Diophantine方程x^3-1=Dy^2的初等解法至今仍未解决.主要利用同余式、平方剩余、Pell方程的解的性质、递归序列,证明了q≡7(mod 24)为奇素数.(q/13)=-1时,Diophantine方程x^3-1=13qy^2仅有整数解(x,y)=(1,0).
