Abstract: R. Hofer and A. Winterhof proved that the 2-adic complexity of the two-prime (binary) generator of period pq with two odd primes $$p e q$$ is close to its period and it can attain the maximum in many cases. When the two-prime generator is applied to producing quaternary sequences, we need to determine the 4-adic complexity. It is proved that there are only two possible values of the 4-adic complexity for the two-prime quaternary generator, which are at least $$pq-1-\max \{\log _4(pq^2),\log _4(p^2q)\}$$ . Examples for primes p and q with $$5\le p, q <10000$$ illustrate that the 4-adic complexity only takes one value larger than $$pq-\max \{\log _4(p),\log _4(q)\}$$ , which is close to its period. So it is good enough to resist the attack of the rational approximation algorithm.

Abstract: In this paper, we discovered a new sequence of all prime numbers which end with a 1 or 9, the sequence defined by 𝑎(𝑛) = (𝑛^2 − 𝑛 − 1)/gcd (𝑏(𝑛), 𝑛^2 − 𝑛 − 1), with 𝑏(𝑛) satisfying the recurcive formula 𝑏(𝑛)=(𝑛 − 1). 𝑏(𝑛 − 1) − 𝑛. 𝑏(𝑛 − 2) and 𝑏(1) = 𝑏(2) = −1, the sequence 𝑎(𝑛) takes only 1’s and primes.

Abstract: Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p mid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The paper contains an explicit formula for the discriminant of $K$ involving only the prime powers dividing $a,n$.

Abstract: We show that every sufficiently large $x\equiv 3(4)$ can be written as the sum of three primes, each of which is a sum of a square and a prime square. The main tools are a transference version of the circle method and various sieve related ideas.

Abstract: The prime-number-formula at any distance from the origin has a systematic error, proportional to the square of the number of primes up to the square root of the distance. The proposed completion in the present paper eliminates by a quickly converging recursive formula the systematic error. The remaining error is reduced to a symmetric dispersion, with standard deviation proportional to the number of primes at the square root of the distance. 1: Evaluation of the number of primes The total number of the primes is the integral of the local logarithmic density of free positions, evaluated by Riemann. The first approximation of the integral is the sum of the logarithmic density over all integers, in the following used as sum over all integers: π c ( )= 2 c c 1 ln c ( )     d πln_appr c ( )  2 c n 1 ln n ( )   (1.1) This above sum may be written as summing up first over all integers within the sections of the length ( c ) and then summing up over all the ( c ) sections of the length ( c ). Taking the average value over each section and summing up over the sections is an approximation, in the following used as sum over all sections.

Abstract: A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: MN = 2N – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0; MN]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.

Abstract: • An analytic evolutionary game theory model involving the prisoner's dilemma is shown to identify the prime numbers. • The model confirms a longstanding hypothesis positing a connection between prime numbers and the cross-disciplinary puzzle of how cooperation evolved. • The model provides a theoretical foundation for future research into strategies for cooperation and free-riding when social interaction occurs across discrete time points with varying social partners. The development of methods to identify prime numbers spans centuries and includes models of physical and biological systems that spot primes. This paper adds to the latter research genre by reporting a prisoner's dilemma model that identifies prime numbers greater than 2. Albeit containing unconventional features and arguable assumptions, the model nonetheless confirms a previously hypothesized connection between prime numbers and the cross-disciplinary puzzle of how cooperation evolved. In a companion paper (part II), the features and assumptions of the analytic model reported here are explored in a finite-population, computational model.

Abstract: Let $K=\mathbb {Q}(\sqrt [n]{a})$ be an extension of degree $n$ of the field $\mathbb Q $ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p mid a$ or the highest power of $p$ dividing $a$ is coprime to $p$

Abstract: In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P1 = 1 + 6 * n (n > 0, n is an integer) P5 = 5 + 6 * m (m ≥ 0, m is an integer) Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Twin prime numbers can only be formed between P1 and P5 prime numbers, n must be 1 greater than m. The largest prime number is known at the moment 2^(82,589,933) – 1 is a P1 prime number.

Abstract: This paper does not claim to prove the Goldbach conjecture, but it does provide a new way of proof (LiKe sequence); And in detailed introduces the proof process of this method: by indirect transformation, Goldbach conjecture is transformed to prove that, for any odd prime sequence (3, 5, 7, …, Pn), there must have no LiKe sequence when the terms must be less than 3 × Pn. This method only studies prime numbers and corresponding composite numbers, replaced the relationship between even numbers and indeterminate prime numbers. In order to illustrate the importance of the idea of transforming the addition problem into the multiplication problem, we take the twin prime conjecture as an example and know there must exist twin primes in the interval [3Pn, P2n]. This idea is very important for the study of Goldbach conjecture and twin prime conjecture. It’s worth further study.

Abstract: Summary This paper reports on the formalization of ten selected problems from W. Sierpinski’s book “250 Problems in Elementary Number Theory” [5] using the Mizar system [4], [1], [2]. Problems 12, 13, 31, 32, 33, 35 and 40 belong to the chapter devoted to the divisibility of numbers, problem 47 concerns relatively prime numbers, whereas problems 76 and 79 are taken from the chapter on prime and composite numbers.

Abstract: We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^{\theta})$. We obtain in particular \[ N(\alpha, T) \ll T^{\frac{c(1-\alpha)}{1-\theta}}\log^{9} T, \] for a constant $c$ arbitrarily close to $4$, improving significantly the current state of the art. We also investigate the consequences of the obtained zero-density estimates on the PNT in short intervals. Our proofs crucially rely on an extension of the classical mean-value theorem for Dirichlet polynomials to generalized Dirichlet polynomials.

Abstract: Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field $F'$ such that $E$ has semistable reduction at every prime of $F'$ above $p$ and is supersingular at at least one prime above $p$. Under appropriate hypotheses, we compute the Akashi series of the signed Selmer groups of $E$ over a $\mathbb{Z}_p^d$-extension over a finite extension $F$ of $F'$. As a by-product, we also compute the Euler characteristics of these Selmer groups.

Abstract: • The paper reports a model of a finite population of agents constrained to strategies that alternate between activity and inactivity (a.k.a. temporal partitioning) in a social environment where multiple one-shot prisoner's dilemma games occur across discrete, intra-generational time points. • Numerical simulation of the model indicates that cooperators reach fixation with far greater frequency when using schedules with prime-number period lengths. • Simulation results dovetail with the findings of a recent analytic model that confirmed a longstanding, hypothesized link between the prime numbers and the evolution of cooperation. • The findings suggest that schedules with prime-number period lengths constitute a new mechanism for the evolution of cooperation. • Viewed in concert with the findings of past-predator prey models, the results raise the possibility that cyclical behavior with prime-number period lengths might serve as an adaptive solution to a range of challenges that lifeforms face. This paper presents a model of a finite population of agents constrained to strategies that alternate between activity and inactivity (a.k.a. temporal partitioning) in a social environment where multiple one-shot prisoner's dilemma games occur across discrete, intra-generational time points. Evolutionary selection acts on agents’ behavioral dispositions to cooperate/defect and the schedules that determine when agents periodically implement that behavior. Numerical simulation of the model indicates that cooperators reach fixation with far greater frequency when using schedules with prime-number period lengths. These findings reinforce recent analytic findings that indicate a connection between the evolution of cooperation and the prime numbers, plus they offer new empirical predictions about the timing of social behavior.

Abstract: Purpose: Primes are notorious for their irregular distribution in natural numbers. Such a lack of regularity makes primes elusive. Many NP-hard problems are related to the irregular occurrence of primes in natural numbers. Methods: To extract the underlying regularity of prime distribution, author started from the complementary relationship between composites and primes, through the regular occurrence of composites to infer the regularity underlying primes. Results: Previously random-appearing occurrence of primes resulted from the regular periodic decimations of various frequencies and cycles set by primes. Conclusions: Primes are the survivors of natural numbers after periodic decimations caused by primes. This leads to a novel concise representation of the set of all primes using sine function, suggestive of periodicity for both primes and composites.

Abstract: We consider non-monogenic simplest cubic fields $K=\mathbb{Q}(\rho)$ in the family introduced by Shanks, and among these, we focus in the fields whose generalized module index $[\mathcal{O}_K:\mathbb{Z}[\rho]]$ is a prime number $p$. We prove that these fields arise exactly for $p=3$ or $p\equiv1\,(\mathrm{mod}\,6)$ and we use the method introduced in arXiv:2005.12312 to find the additive indecomposables of $\mathcal{O}_K$. We determine the whole structure of indecomposables for the family with $p=3$ and obtain that the behaviour is not uniform with respect to the indecomposables of $\mathbb{Z}[\rho]$. From the knowledge of the indecomposables we derive some arithmetical information on $K$, namely: the smallest and largest norms of indecomposables, the Pythagoras number of $\mathcal{O}_K$ and bounds for the minimal rank of universal quadratic forms over $K$.

Abstract: RSA algorithm is an asymmetric public key encryption algorithm, which is widely used in data encryption and digital signature. Based on the analysis of RSA algorithm and Miller Rabin Prime number decision algorithm, the prime number decision algorithm is improved, and the experimental results show that the improved algorithm improves the generation efficiency of large prime number in RSA algorithm, so as to reduce the encryption and decryption time of RSA algorithm.

Abstract: We prove explicit versions of the Kronecker–Weyl theorems, both in a discrete and a continuous settings, without any linear independence hypothesis. As an application, we propose an alternative approach to problems concerning asymptotic densities in prime number races, over number fields and over function fields in one variable over finite fields, in the language of random variables. Our approach allows us to prove new results on the existence and positivity of some of those densities, which, in the case of races over function fields, do not require any linear independence hypothesis.

Abstract: The distribution of twin prime numbers is discussed. The research method of corresponding prime number distribution is proposed. The distribution of prime numbers corresponding to integers and composite numbers is discussed. Through the corresponding prime distribution rate of integers and composite numbers, it is found that the corresponding prime distribution rate of composite numbers approaches the corresponding prime distribution rate of integers. The distribution principle of corresponding prime number of composite number is proved. The twin prime distribution theorem is obtained. The number of twin prime numbers is thus obtained. It provides a practical way to study the conjecture of twin prime numbers.

Abstract: This paper is concerned with the relationship of $y$-friable (i.e. $y$-smooth) integers and the Dickman function. Under the Riemann Hypothesis (RH), an asymptotic formula for the count of $y$-friable integers up to $x$, $\Psi(x,y)$, in terms of the Dickman function was previously available only for $y \ge (\log x)^{2+\varepsilon}$, by works of Hildebrand and Saias. Unconditionally we establish an asymptotic formula for $\Psi(x,y)$ in the wider range $y \ge (1+\varepsilon)\log x$, whose shape is $x \rho(\log x/\log y)$ times correction factors. These factors take into account the contributions of zeta zeros and prime powers. With this formula at hand, we resolve two questions of Hildebrand and Pomerance. Hildebrand conjectured that $\Psi(x,y)$ is not $\asymp x \rho(\log x/\log y)$ once $y$ is smaller than $(\log x)^{2+\varepsilon}$, and we show unconditionally he was correct. Pomerance asked whether the inequality $\Psi(x,y)\ge x \rho(\log x/ \log y)$ holds for $x/2 \ge y \ge 2$. If RH is false we show this fails infinitely often. When RH is true, the inequality holds for $x/2\ge y\ge 2$, $x \gg 1$ except possibly for $y$ close to the critical point $y=(\log x)^2$. Near this point, the question is essentially equivalent to \[ \liminf_{y \to \infty} \frac{\psi(y)-y}{\log y \sqrt{y} } > L\] for a constant $L\approx -0.666217$, where $\psi$ is the Chebyshev function. It is expected that this limit is $0$, but even under RH we cannot rule out that it is $-\infty$. As another consequence of our formula, we show that under RH $\Psi(x,y)$ exhibits an unexpected phase transition when $y \approx (\log x)^{3/2}$. We also improve old inequalities for $\Psi(x,y)$.

Abstract: A wave function constructed from prime counting function is employed to study the properties of primes using quantum dynamics. The prime gaps are calculated from the expectation values of position and a formula for maximal gaps is proposed. In an analogous nonlinear system, the trajectories, associated nodes with their stability condition and the bifurcation dynamics are studied using classical dynamics. It is interesting to note that the Lambert W functions appear as a natural solution for the fixed points as functions of energy. The derived potential with the divergence resembles the effective potential experienced by a particle near a massive spherical object in general theory of relativity. The coordinate time and proper time corresponding to a black hole serendipitously find their analogy in the solution of the nonlinear dynamics representing primes. The stereographic projection obtained from quantum dynamics on unit circle in the $$(\theta ,p_{\theta })$$ phase space of the real numbers present along x-axis in general and prime numbers in particular provides a simple way to calculate a formula for upper bounds on the prime gaps. The estimated prime gaps is found to be significantly better than that of Cramer’s predicted values.

Abstract: After shocking the mathematics community with a major result in 2013, Yitang Zhang now says he has solved an analogue of the celebrated Riemann hypothesis. After shocking the mathematics community with a major result in 2013, Yitang Zhang now says he has solved an analogue of the celebrated Riemann hypothesis.

Abstract: N continuous prime numbers can combine a group of continuous even numbers. If an adjacent prime number is followed, the even number will continue. For example, if we take the prime number 3, we can get an even number 6. If we follow an adjacent prime number 5, we can get even numbers by using 3 and 5: 6, 8 and 10. If a group of continuous prime numbers 3, 5, 7, 11, ..., P, we can get a group of continuous even numbers 6, 8, 10, 12, 2n. Then if an adjacent prime number q is followed, the Original group of even numbers 6, 8, 10, 12, 2n will be finitely extended to 2(n + 1) or more adjacent even numbers. My purpose is to prove that the continuity of prime numbers will lead to even continuity as long as 2(n + 1) can be extended. If the continuity of even numbers is Discontinuous, it violates the Bertrand Chebyshev theorem of prime Numbers. Because there are infinitely many prime numbers: 3, 5, 7, 11, We can get infinitely many continuous even numbers: 6, 8, 10, 12,Get: Gold Bach conjecture holds. 2020 Mathématiques Subjectif Classification: 11P32, 11U05, 11N05, 11P70. Research ideas: If the prime number is continuous and any pairwise addition can obtain even number continuity, then Gold Bach’s conjecture is true. Human even number experiments all get (prime number + prime number). I propose a new topic: the continuity of prime numbers can lead to even continuity. I designed a continuous combination of prime numbers and got even continuity. If the prime numbers are combined continuously and the even numbers are forced to be discontinuous, a breakpoint occurs. It violates Bertrand Chebyshev's theorem. It is proved that prime numbers are continuous and even numbers are continuous. The logic is: if Gold Bach's conjecture holds, it must be that the continuity of prime numbers can lead to the continuity of even numbers. Image interpretation: turn Gold Bach’s conjecture into a ball, and I kick the ball into Gold Bach’s conjecture channel. There are several paths in this channel and the ball is not allowed to meet Gold Bach’s conjecture conclusion in each path. This makes the ball crazy, and the crazy ball must violate Bertrand Chebyshev's theorem.

Abstract: The objective of this short paper is to give and prove a main theorem confirming that any interval of the special form: ]an+k + Pn, an+k+1 + Pn[ (n 2 N) does not contain any primes, for all k 2 N such that an+k+1 < a2 n+1 (an is the nth prime number & Pn is the nth prime factorial). Then we give several counterexamples of such intervals, which contain primes, when the condition (an+k+1 < a2 n+1) is not satisfied.

Abstract: Aiming at the exposure of the public key in the RSA algorithm, this paper proposes to use the double RSA algorithm to perform the second encryption of the public key modulus to generate a false modulus, thereby improving the security of the RSA algorithm. Secondly, the problem of generating large prime numbers is a time-consuming problem in the RSA algorithm. In addition to generating large prime numbers in encrypted plaintext, double RSA algorithm also generates large prime numbers that create fake modulus, which undoubtedly increases the time-consuming algorithm. Aiming at the above-mentioned time-consuming problem, this paper proposes an improved random incremental search algorithm, which uses low-order four prime numbers instead of traditional high-order two prime numbers. Experimental data shows that the improved algorithm improves the running speed and has certain feasibility.

Abstract: The class number \(h_K\) of an algebraic number field K plays an important role in the arithmetic of K. One would like to have an explicit formula for \(h_K\) in terms of simpler values depending upon the field K. Since all ideals of \({{\mathcal {O}_K}}\) are product of prime ideals and the number of prime ideals of \({{\mathcal {O}_K}}\) is infinite, to compute \(h_K\) in finite number of steps, one has to use some infinite processes, e.g., infinite series, infinte products and some analytic concepts as has been done in the present chapter to prove Dirichlet’s Class Number Formula.