Probability, information theory, and prime number theory

Clustering of a Number of Genes Affecting in Milk Production using Information Theory and Mutual Information

Information theory is a branch of mathematics. Information theory is used in genetic and bioinformatics analyses and can be used for many analyses related to the biological structures and sequences. Bio-computational grouping of genes facilitates genetic analysis, sequencing and structural-based analyses. In this study, after retrieving gene and exon DNA sequences affecting milk yield in dairy ...

Three Topics in Additive Prime Number Theory

We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectu...

Some Topics in Prime Number Theory

(ii) p has no divisor other than 1 and itself We denote by P the set of all primes and we write the elements of P in the increasing order as p 1 < p 2 < p 3 < · · · Now consider = p 1 · · · p n + 1. Observe that is an integer > 1. Therefore there exists q ∈ P with q |. Then q = p i for 1 ≤ i ≤ n since q || − 1. Thus q > p n. We started with primes p 1 , · · · , p n and we found a prime q > p n....

New results in comparative prime number theory

Logical basis for information theory and probability theory

A new logical basis for information theory as well as probability theory is proposed, based on computing complexity.