Novel Aspects Of The Global Regularity Of Primes

https://econteenblog.wordpress.com/, 2018

One thing that will be investigated is if there can be two prime number ratios that are equal to each other. There will be other aspects of prime numbers that will be investigated as well, such as how they ‘relate’ to composite numbers. This paper was originally intended to show that it is impossible to find an underlying pattern or explanation by just using natural numbers, but that was likely incorrect.

paper, 2020

Prime numbers are of greater interest to mathematicians, both professional and amateur, since people began to study the properties of numbers and find them fascinating. On the one hand, prime numbers seem to be randomly distributed among natural numbers with no law other than probability. On the other hand, however, the distribution of primes globally reveals remarkably smooth regularity when viewed in the context of their products. This can be described by the formula π(N) + ∑p(p ’) = ½N, which says that half of a given quantity is the sum of the number of primes to a given quantity and their products. The combination of the number of prime numbers π(N) with their products greater than 3 ∑p(p')> 3 always creates a constant value growing in progress 34+1(q), and their products of the number 3 ∑3(p) in progress 17-1(q), and half of a given quantity of ½N progressively 51(q). (34 + 1)q + (17-1)q = (51)q, [π(N) + ∑p(p ')> 3] + ∑3(p) = ½N, (26+9) + 16 = (34+1)+(17-1) = 51

We present some new ideas on important problems related to primes. The topics of our discussion are: simple formulae for primes, twin primes, Sophie Germain primes, prime tuples less than or equal to a predefined number, and their infinitude; establishment of a kind of similarity between natural numbers and numbers that appear in an arithmetic progression, similar formulae for primes and the so called generalized twin primes in an arithmetic progression and their infinitude; generalization of Bertrand postulate and a Bertrand like postulate for twin primes; some elementary implications of a simple primality test, the use of Chinese remainder theorem in a possible proof of the Goldbach conjecture; Schinzel Sierpinski conjecture; and lastly the Mersenne primes and composites, Fermat primes, and their infinitude.

2021

The paper is the ultimate prime numbers algorithm that gets rid of the unneccessary mystery about prime numbers. All the numerous arithmetic series patterns observed between various prime numbers are clearly explained with an elegant "pattern of remainders". With this algorithm we prove that odd numbers too can make an Ulam spiral contrary to current ""proofs". At the end of the paper this author proves the relationship between a simple arithmetic series pattern and the Riehmann's prime numbers distribution equation. This paper would be important for encryption too. As an example, prime integer 1979 is expressed as 1.2.4.5.10.3.7.3.1.7.26.18.11.1. This makes even smaller primes useful for encryption as well.

2017

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples. Keywords prime number generator, prime number theorem, twin primes, erratic nature of primes

Advances in Pure Mathematics, 2014

The main design of this paper is to determine once and for all the true nature and status of the sequence of the prime numbers, or primes-that is,

Turkish Journal of Analysis and Number Theory, 2015

The purpose of this paper is to introduce a new pattern in Primes numbers, to eliminate the randomness in their patterns. This paper also justifies the solutions in a numerical and geometric manner. The Prime Function provides further distinction in the nature of Prime Numbers by distinguishing the nature of normality and Abnormality in Prime Numbers. To verify the normality of corresponding Prime numbers, the Sufi primality test is formed. Also using the Prime Function, the formula for the approximate sum of Prime Numbers is derived. The limitations and conditions of the Prime function are also stated. These factors provide a panoramic view of the Prime Function and its potential factor in Number Theory [2].

2017

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples.

The primes, including the twin primes and the other prime pairs, are the building-blocks of the integers. Euclid's proof of the infinitude of the primes is generally regarded as elegant. It is a proof by contradiction, or, reductio ad absurdum, and it relies on an algorithm which always brings in larger and larger primes, an infinite number of them. However, the proof is also subtle and is misinterpreted by some, with one well-known mathematician even remarking that the algorithm might not work for extremely large numbers. A long unsettled related problem, the twin primes conjecture, has also aroused the interest of many researchers. The author has been conducting research on the twin primes for a long time and had published a paper on them (see, B. Wong, Possible solutions for the "twin" primes conjecture - The infinity of the twin primes, International Mathematical Journal, 3(8), 2003, 873-886). This informative paper presents some important facts on the twin primes which would be of interest to prime number researchers, with some remarks/reasons that point to the infinitude of the twin primes, including a reasoning which is somewhat similar to Euclid's proof of the infinity of the primes; very importantly, two algorithms are developed in Section 5 for sieving out the twin primes from the infinite list of the integers, which would be of interest to cryptographers and even to computer programmers. [Published in an international mathematics journal. Thoroughly edited & re-arranged by the Editor-in-Chief of the journal.]

2022

For any m=3(2n+1), n≥1, the prime counting function π(m)=4+|A4(m)|+2|A6(m)| where A6(m) and A4(m) are the sets of Twin Primes and "Isolated" Primes, below m, respectively. T(m)=1+|A2(m)|+|A4(m)|+|A6(m)| is the number of consecutive composite odd numbers (COCOONs) below m where A2(m) is the set of pairs of COCOONs, below m and distant by 2. With m odd, π(m), T(m) and |A2(m)|=m−92−3|A6(m)|−2|A4(m)| yield 4|A6(m)|+7=m−2(T(m)+|A4(m)|). Hence 0<1−2m(T(m)+|A4(m)|)<1. Thus Sm≡{1−2m(T(m)+|A4(m)|), m as above}⊂ℝ is bounded. Therefore Inf(Sm)≥0 exists, is unique and finite. We then introduce α=Inf(Sm) if Inf(Sm)>0 and α=ϵ if Inf(Sm)=0, with 0<ϵ<1−2m(T(m)+|A4(m)|). ℚ Dense in ℝ guarantees the existence of ϵ>0, in ℚ. In any case, 0<α<1−2m(T(m)+|A4(m)|). Therefore, 4|A6(m)|+7>αm. Hence limm⟶∞|2A6(m)|=+∞. Similarly, limm⟶∞|A4(m)|=+∞.