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This study suggests grouping of numbers that do not divide the number 3 and/or 5 in eight columns. Allocation results obtained from multiplication of numbers is based on column belonging to him. If in the Sieve of Eratosthenes the majority of multiplication of prime numbers result in a results devoid of practical benefit (numbers divisible by 2, 3 and/or 5), in the sieve of prime numbers using algorithms, each multiplication of prime number gives a result in a number not divisible to 2, 3 and/or 5.

In the [16] has been proven that the multiplicative group Z/pZ∗ is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group. Meanwhile, in cryptographic system like RSA, in which security basis depends upon the difficulty of factorization of given numbers into prime factors, it is important to employ integers that are difficult to be factorized into prime factors. If both p and 2p + 1 are prime numbers, we call p as Sophie Germain prime, and 2p+ 1 as safe prime. It is known that the product of two safe primes is a composite number that is difficult for some factoring algorithms to factorize into prime factors. In addition, safe primes are also important in cryptography system because of their use in discrete logarithm based techniques like DiffieHellman key exchange. If p is a safe prime, the multiplicative group of numbers modulo p has a subgroup of large prime order. However, no definitions have not been established yet with the safe prime and Sophie Germain prime. So it is important to give definitions of the Sophie Germain prime and safe prime. In this article, we prove finite subgroup of the multiplicative group of a field is a cyclic group, and, further, define the safe prime and Sophie Germain prime, and prove several facts about them. In addition, we define Mersenne number (Mn), and some facts about Mersenne numbers and prime numbers are proven.

We report the factorization of a 135-digit integer by the triple-large-prime variation of the multiple polynomial quadratic sieve. Previous workers [6][10] had suggested that using more than two large primes would be counterproductive, because of the greatly increased number of false reports from the sievers. We provide evidence that, for this number and our implementation, using three large primes is approximately 1.7 times as fast as using only two. The gain in efficiency comes from a sudden growth in the number of cycles arising from relations which contain three large primes. This effect, which more than compensates for the false reports, was not anticipated by the authors of [6] [10] but has become quite familiar from factorizations obtained using the number field sieve. We characterize the various types of cycles present, and give a semi-quantitative description of their rather mysterious behaviour.

Surprisingly, the class numbers of cyclotomic fields have only been determined for fields of small conductor, e.g. for prime conductors up to 67, due to the problem of finding the "plus part," i.e. the class number of the maximal real subfield. Our results have improved the situation. We prove that the plus part of the class number is 1 for prime conductors between 71 and 151. Also, under the assumption of the generalized Riemann hypothesis, we determine the class number for prime conductors between 167 and 241. This technique generalizes to any totally real field of moderately large discriminant, allowing us to confront a large class of number fields not previously treatable., Accepted by Mathematics of Computation. 12 pages

In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. We then revisit Thorne’s applications and give a better bound in each case. Keywords: bounded prime gaps; square-free numbers; modular elliptic curves

International audience; When $K \ge 1$ is an integer and $S$ is a set of prime numbers in the interval $(N/2 , N ]$ with $|S| \ge\pi^* (N)/K$, where $\pi^* (N)$ is the number of primes in this interval, we obtain an upper bound for the additive energy of $S$, which is the number of quadruples $(x_1 , x_2 , x_3 , x_4)$ in $S^4$ satisfying $x_1 + x_2 = x_3 + x_4$. We obtain this bound by a variant of a method of Ramaré and I. Ruzsa. Taken together with an argument due to N. Hegyvári and F. Hennecart this bound implies that when the sequence of prime numbers is coloured with $K$ colours, every sufficiently large integer can be written as a sum of no more than $CK \log \log 4K$ prime numbers, all of the same colour, where $C$ is an absolute constant. This assertion is optimal upto the value of C and answers a question of A. Sárközy.

Let $a\in \mathbb{Z}\setminus \{0\}$. A positive squarefree integer $N$ is said to be an $a$-Korselt number ($K_{a}$-number, for short) if $N\neq a$ and $p-a$ divides $N-a$ for each prime divisor $p$ of $N$. By an $a$-Williams number ($W_{a}$-number, for short) we mean a positive integer which is both an $a$-Korselt number and $(-a)$-Korselt number. This paper proves that for each $a$ there are only finitely many $W_{a}$-numbers with exactly three prime factors, as conjectured in 2010 by Bouallegue-Echi-Pinch.

At present noone knows an unconditional proof that between two consecutive squares there is always a prime number. In a previous paper the author proved that, under the assumption of the Lindelof hypothesis, each of the intervals [n 2, (n+1)2] ⊂ [1,N], with at most O(N ɛ) exceptions, contains the expected number of primes, for every constant ɛ > 0. In this paper we improve the result by weakening the hypothesis in two different ways.

Let p be an odd prime number and ℓ a prime number with ℓ ≠ p. Let Kn = Q(ζpn+1) be the pn+1-st cyclotomic field. Let hn and hn- be the class number and the relative class number of Kn, respectively. When ℓ = 2, we give an explicit bound mp depending on p such that the ratio hn-/hn-1- is odd for all n > mp. When ℓ ≥ 3, we also give a corresponding result on the ℓ-part of the relative class number of Kn+(ζℓ). As an application, we show that when p ≤ 509, the ratio hn/h0 is odd for all n ≥ 1.

Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta$ such that any preassigned sequence of $k$ digits occurs in the $q$-ary expansion of $\eta$ at the expected frequency, namely$1/q^k$. In a series of recent papers, using the complexity of the multiplicative structure of integers along with a method we developed in 1995 regarding the distribution of subsets of primes in the prime factorization of integers, we initiated new methods allowing for the creation of large families of normal numbers. Here, we further expand on this initiative.

Assuming the Riemann hypothesis, we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Ω-term, we show that our result is essentially the best possible.

Lagrange proved that every positive integer is the sum of four squares of natural numbers. Although Lagrange’s proof is constructive, it is not known whether the relative algorithm produces a four-square representation of any prime or non-prime integer with deterministic polynomial complexity. Limited to prime numbers, it is proved that their representation as the sums of four squares can be obtained with deterministic polynomial complexity. The key to this computational accomplishment is the evaluation of square roots modulo a prime, through Schoof’s method for counting the number of points on elliptic curves over prime fields.

We studied the distribution of units of an algebraic number field modulo prime ideals. Here we study the distribution of units of a cubic abelian field modulo rational prime numbers. For a decomposable prime number $p$ , $2(p-1)^2$ is an upper bound of the order of the unit group modulo $p$ , and we show that the conjectural density of primes which attain it is really positive.

We study the distribution of prime numbers that have a given number of one bits in their binary representation, and of those that have a given number of zero bits. We consider basic questions such as whether there are infinitely many of them, and explain their distribution in residue classes modulo small primes. We prove the unexpected result that, for m ≥ 3, there is no prime number with precisely 2m bits, exactly two of which are zero bits.

We derive a formula for the density of positive integers satisfying a certain system of inequality, often referred as prime number races, in the case of the polynomial rings over finite fields. This is a function field analog of the work of Feuerverger and Martin, who established such formula in the number field case, building up on the fundamental work of Rubinstein and Sarnak.

Recent results of the author are used to permit applications of the sieve method of Selberg and others to problems involving binary forms with integer coefficients. Results are obtained showing that infinitely many of the numbers represented by suitable cubic forms (with integer values of the arguments) have a prime factor that is fairly large when compared with the number itself. It is also shown that infinitely many of the numbers represented by suitable forms of degree three or more have a bounded number of prime factors. For cubic forms the bound obtained is two.

In this paper, we prove: "Suppose α is an irrational number and let || y || denote the smallest distance of y from an integer. Then, for any real number β, there are infinitely many primes p such that ||α p − β|| p −4/13 ."

Suppose that h is a positive integer. For an integer n ≥ 2 , define P h ( n ) = ( 1 ω ( n ) ∑ p ∣ n p prime p h ) 1 / h , where ω ( n ) denotes the number of distinct prime divisors of n . Let A h ( x ) be the set of all positive integers n ≤ x with ω ( n ) > 1 such that P h ( n ) is prime and P h ( n ) ∣ n . In this paper, we prove that x exp ( 2 h log x log log x ) ≤ | A h ( x ) | ≤ x exp ( ( 1 / 2 ) log x log log x ) , which generalizes a result of Luca and Pappalardi for h = 1 .

For a given positive integer m and an algebraic number field K necessary and sufficient conditions for the mth cyclotomic polynomial to have K-integral solutions modulo a given integer of K are given. Among applications thereof are: that the solvability of the cyclotomic polynomial mod an integer yields information about the class number of related number fields; and about representation of integers by binary quadratic forms. The latter extends previous work of the author. Moreover some information is obtained pertaining to when an integer of K is the norm of an integer in a given quadratic extension of K. Finally an explicit determination of the pqth cyclotomic polynomial for distinct primes p and q is provided, and known results in the literature as well as generalizations thereof are obtained.

Let p be an odd regular prime number. We prove that there exist infinitely many totally real number fields k of degree p −1 whose class numbers are not divisible by p . Moreover, for certain regular prime number p , we prove that there exist infinitely many totally real number fields k of degree p −1 whose Iwasawa λ p -, μ p -invariants vanish.

We employ a type number formula from the theory of quaternion algebras to gain information on the 2-part of the class numbers of imaginary quadratic number fields whose discriminants are divisible by three or fewer prime numbers.

It is proved that every sufficiently large odd integer n can be written as n=x+p13+p23+p33+p43 where p1, p2, p3, p4 are primes, and x has at most two prime factors.

We found the asymptotics, p→∞, for the number of cycles for iteration of monomial functions in the fields of p-adic numbers. This asymptotics is closely connected with classical results on the distribution of prime numbers.

We study sets P of k primes that satisfy the condition gcd(∏ A −∏ B , ∏ P )=1 whenever A and B are disjoint non-empty subsets of P . It is known that such sets of primes exist for all positive integers k . It is of interest to know the asymptotic behavior of n k , the smallest natural number that is the product of k such primes. In this paper we derive asymptotic bounds for n k .

Without using the prime number theorem, we obtain the asymptotics of the $r$th largest prime divisor of a harmonically distributed random positive integer $N$; harmonic asymptotics are obtained from asymptotics of the zeta distribution via Tauberian methods. (Knuth and Trabb-Pardo need a strong form of the prime number theorem to obtain the distributions when $N$ is uniformly distributed.) A trick brings in Poisson variates, and then we can use the methods developed for the fractional length of the $r$th longest cycle in a random permutation.

We prove results on the asymptotic behavior of large prime factors of the integers. The basic idea of the paper is that if k is any fixed integer, then the kth largest prime factor of n, denoted by P,(lz) is generally much bigger than CjJ’Jn). We give precise estimates of this phenomenon. This paper is a sequel to an earlier paper by the authors. 1. Notations and definitions‘ Throughout this paper the letters p and q, with or without subscript will denote primes. Let 12 = IIrE1 p?, p, > p, > a.. > p, be the canonical decomposition of an integer n > 1 into primes. We set

We give the complete solutions of the equations 2*3^ + 1 = 2 + 3, 2V + 2 = 3 + 1 and 2*3' + 3* = 2 + 1 in integers x, y, z, w. We use this to prove that every large rational number has at most four representations of the form 23^ + 2 + 3^. Finally we prove that, for given integer n and prime numbers p\,..., pt9 every rational number m has at most C representations of the form Σ" = 1 p\ ύ pkt u where ki\,...,kn are integers.

Throughout this note,pdenotes a fixed prime number andfdenotes a fixed natural number prime top.It is easy to see and more or less known that for any natural numbern, there exists an elliptic curve overpwhosej-invariant is of degreenoverFpand whose endomorphism ring is isomorphic to an order of an imaginary quadratic field. In this note, we consider a more precise problem:for any natural number n, decide whether or not there exists an elliptic curve overpwhose j-invariant is of degree n over Fpand whose endomorphism ring is isomorphic to an order of an imaginary quadratic field with conductor f.

has a solution g that is unique modulo m. The exponent d of g modulo m is the least common multiple of 4(pl), ..., 0(p,), where sb(n) is the Euler function. In this paper we consider the problem of determining the number N(g) of solutions in s, t of the congruence gS + 1 gt (mod m), where the values of s and t are each selected from the integers 0, 1, . ., d 1. We discuss for the most part the case in which m is the product of two distinct odd primes p and q. If e denotes the greatest common divisor of p 1 and q 1, then d = (p 1)(q 1)/e. Let g denote a common primitive root of p and q, and let N(g) = Ne(g) denote the number of solutions in s,t of the congruence gs + 1 gt (mod pq), where 0 < s, t < d 1. In [3] the function Ne(g) is required for the construction of certain types of residue difference sets. Indeed, it is this application of Ne(g) that provided the motivation for the present paper (see the corollary of Theorem 3 in ?5). The case e = 2 is treated in ?4. We find that