On the order of the non-primes of type 6n+5 and 6n+1

The fundamental importance of prime numbers for mathematics is that all other natural numbers can be represented as a unique product of prime numbers[1]. For centuries, mathematicians have been trying to find an order in the occurrence of prime numbers. Since Riemann´s paper on the number of primes under a given size, the distribution of the primes is assumed to be random[2]. Here, I show that prime numbers are not randomly distributed using three parametric sine functions. Their positions are determined by the order of occurrence of the nonprimes of types 6n+5 and 6n+1. Furthermore, I will show an exact primality test using these three parametric sine functions.

On the order of the non-primes of type 6n+5 and 6n+1

The fundamental importance of prime numbers for mathematics is that all other natural numbers can be represented as a unique product of prime numbers[1]. For centuries, mathematicians have been trying to find an order in the occurrence of prime numbers. Since Riemann´s paper on the number of primes under a given size, the distribution of the primes is assumed to be random[2]. Here, I show that prime numbers are not randomly distributed. Their positions are determined by the order of occurrence of the nonprimes of types 6n+5 and 6n+1. By using parametric sine functions, I will show here that all assumptions known today regarding the distribution of prime numbers larger than 3 are wrong. In particular, this refutes Riemann´s hypothesis of random distribution of prime numbers. Furthermore, I will show an exact primality test based on these parametric sine functions, which only uses one parameter.

On the order of the non-primes of type 6n+5 and 6n+1

The fundamental importance of prime numbers for mathematics is that all other natural numbers can be represented as a unique product of prime numbers[1]. For centuries, mathematicians have been trying to find an order in the occurrence of prime numbers. Since Riemann´s paper on the number of primes under a given size, the distribution of the primes is assumed to be random[2]. Here, I show that prime numbers are not randomly distributed. Their positions are determined by the order of occurrence of the nonprimes of types 6n+5 and 6n+1. By using parametric sine functions, I will show here that all assumptions known today regarding the distribution of prime numbers larger than 3 are wrong. In particular, this refutes Riemann´s hypothesis of random distribution of prime numbers. Furthermore, I will show an exact primality test based on these parametric sine functions, which only needs the calculation modes +, -, · , : and only uses one parameter. There is no such deterministic primality test existing until today[3] [4].

On the P-Adic Valuations of Stirling Numbers of the Second Kind

In this paper, we introduced certain formulas for p-adic valuations of Stirling numbers of the second kind S(n, k) denoted by vp(S(n, k)) for an odd prime p and positive integers k such that n ≥ k. We have obtained the formulas, vp(S(n, n − a)) for a = 1, 2, 3 and vp(S(cpn, cpk )) for 1 ≤ c ≤ p − 1 and primality test of positive integer n. We have presented the results of vp(S(p2, kp)) for 2 ≤ k ≤ p − 1, 2 < p < 100 and a table of vp(S(p, k)). We have posed the following conjectures from our analysis: 1. Let p ≠ 7 be an odd prime and k be an even integer such that 0 < k < p − 1. Then vp(S(p2, kp))-vp(S(p2, p(k+1)) = 3 2. If k be an integer such that 1 < k < p − 1, then the p-adic valuations satisfy vp(S(p2, kp)) = 5 or 6, if k is even; 2 or 3, if k is odd for any prime p > 7. 3. For any primes p and positive integer k such that 2 ≤ k ≤ p − 1, then vp(S( p, k )) ≤ 2.

Combinatorial primality test

In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then ([EQUATION]) and (2) If an integer of the form n = 4k + 3, k ≥ 0 is prime, then [EQUATION]. In addition, the author proposes important conjectures based on the converse of the above theorems which aim to establish primality of n. These conjectures are scrutinized by the given combinatorial primality test algorithm which can also distinguish patterns of prime n whether it is of the form 4k + 1 or 4k + 3.