The present paper gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. First it starts with the density of the primes,...The present paper gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. First it starts with the density of the primes, shortly recapitulates the prime-number-formula and the complete-prime-number-formula, the proof of the set of primes as continuum. The theoretical evaluation is followed in annexes by numerical evaluation of the theoretical results and of different constants, which represent inherent properties of the set of primes.展开更多
The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that...The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that Ω(p+2)≤3 and Ω(p+6)≤6,where Ω(n) denotes the total number of prime divisors of an integer n.We also prove a better conditional result,with the above Ω(p+6)≤6 replaced by Ω(p+6)≤3,under the Elliott-Halberstam conjecture.展开更多
Fudan University, Shanghai 200433, China; Department of Physics, National Central University, Chungli, 3 Taiwan 320, China; Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China. 4 An in...Fudan University, Shanghai 200433, China; Department of Physics, National Central University, Chungli, 3 Taiwan 320, China; Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China. 4 An interactive tool to visualize the K-string composition of long DNA sequences including bacterial complete genomes is described. It is especially useful for ex- ploring short palindromic structures in the sequences. The SeeDNA program runs on Red Hat Linux with GTK+ support. It displays two-dimensional (2D) or one-dimensional (1D) histograms of the K-string distribution of a given sequence and/or its randomized counterpart. It is also capable of showing the di?erence of K-string distributions between two sequences. The C source code using the GTK+ package is freely available.展开更多
文摘The present paper gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. First it starts with the density of the primes, shortly recapitulates the prime-number-formula and the complete-prime-number-formula, the proof of the set of primes as continuum. The theoretical evaluation is followed in annexes by numerical evaluation of the theoretical results and of different constants, which represent inherent properties of the set of primes.
基金supported by the National Key Research and Development Program of China (Grant No. 2021YFA1000700)National Natural Science Foundation of China (Grant No. 12031008)。
文摘The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that Ω(p+2)≤3 and Ω(p+6)≤6,where Ω(n) denotes the total number of prime divisors of an integer n.We also prove a better conditional result,with the above Ω(p+6)≤6 replaced by Ω(p+6)≤3,under the Elliott-Halberstam conjecture.
文摘Fudan University, Shanghai 200433, China; Department of Physics, National Central University, Chungli, 3 Taiwan 320, China; Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China. 4 An interactive tool to visualize the K-string composition of long DNA sequences including bacterial complete genomes is described. It is especially useful for ex- ploring short palindromic structures in the sequences. The SeeDNA program runs on Red Hat Linux with GTK+ support. It displays two-dimensional (2D) or one-dimensional (1D) histograms of the K-string distribution of a given sequence and/or its randomized counterpart. It is also capable of showing the di?erence of K-string distributions between two sequences. The C source code using the GTK+ package is freely available.
