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We describe a reciprocity relation between the prime ideal factorization, and related properties, of certain cyclotomic integers of the type ϕ n ( c − ζ m ) in the cyclotomic field of the m-th roots of unity and that of the symmetrical elements ϕ m ( c − ζ n ) in the cyclotomic field of the n-th roots. Here m and n are two positive integers, ϕ n is the n-th cyclotomic polynomial, ζ m a primitive m-th root of unity, and c a rational integer. In particular, one of these integers is a prime element in one cyclotomic field if and only if its symmetrical counterpart is prime in the other cyclotomic field. More properties are also established for the special class of pairs of cyclotomic integers ( 1 − ζ p ) q − 1 and ( 1 − ζ q ) p − 1 , where p and q are prime numbers.

We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np = (p P — 1)/(p - 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investigation leads to interesting new results about the possible prime factors of Np. For example, we show that if p is an odd positive integer and m is a positive integer and q = 4m 2 p + 1 is prime, then q divides p m2p - 1. Then we explain how this theorem influences the probability that q divides Np.

Let α ∈ ℤ\{0}. A positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and p - α divides N - α for each prime divisor p of N. We are concerned, here, with both a numerical and theoretical study of composite squarefree Korselt numbers. The paper contains two main results. The first one shows that for α ∈ ℤ\{0}, the following properties hold: (i) If α ≤ 1, then each composite squarefree Kα-number has at least three prime factors. (ii) Suppose that α > 1. Let p < q be two prime numbers and N ≔ pq. If N is an α-Korselt number, then p < q ≤ 4α - 3. In particular, there are only finitely many α-Korselt numbers with exactly two prime factors. Let α ∈ ℕ\{0}; by an α-Williams number (Wα-number, for short) we mean a positive integer which is both a Kα-number and a K-α-number. Our second main result shows that if p, 3p - 2, 3p + 2 are all prime, then their product is a (3p)-Williams number.

This survey treats two connected questions in analytic number theory: given a set of natural numbers, one may seek numbers with large prime factors in the set. Alternatively, one searches for smooth numbers in the set. Many examples have been studied: the set of values of a polynomial, the set of integers in a short interval, the set of shifted primes p+a and so on. These are discussed at some length, with references to the literature.

A number is called upper (lower) flat if its shift by +1 ( −1) is a power of 2 times a squarefree number. If the squarefree number is 1 or a single odd prime then the original number is called upper (lower) thin. Upper flat numbers which are primes arise in the study of multi-perfect numbers. Here we show that the lower or upper flat primes have asymptotic density relative to that of the full set of primes given by twice Artin’s constant, that more than 53% of the primes are both lower and upper flat, and that the series of reciprocals of the lower or the upper thin primes converges.

The basis number, b(G) ,of a graph G is defined to be the smallest positive integer k such that G has a k-fold basis for its cycle space. We investigate the basis number of semi-strong product of with a path, a cycle, a star, a wheel and a complete graph.

For an odd prime number p and an integer n ≥ 0, let h n be the class number of the p n+1st cyclotomic field Q($$ \zeta _{p^{n + 1} } $$). It is known that when p = 3 or 5, h n is odd for all n ≥ 0. We prove that the same holds also when p = 7.

We study the solubility of the binary additive equation $[m^c] + [p^c] = n$, where $m$ is an integer, $p$ is a prime number, and $c$ is a fixed real number in the range $1 < c < 3/2$., 10 pages

A prime number p is called elite if only finitely many Fermat numbers 22n + 1 are quadratic residues modulo p. So far, all 21 elite primes less than 250 billion were known, together with 24 larger items. We completed the systematic search up to the range of 2.5 · 1012 finding six more such numbers. Moreover, 42 new elites larger than this bound were found, among which the largest has 374 596 decimal digits. A survey on the knowledge about elite primes together with some open problems and conjectures are presented.

The problem of estimation of the maximal number H of consecutive integer numbers such that they all are either quadratic residues or quadratic nonresidues modulo a prime number p is considered.

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.

Let ω ( n ) \omega (n) be the number of distinct prime factors of n n . For any positive integer k k let n = n k n=n_k be the smallest positive integer such that ω ( n + 1 ) , … , ω ( n + k ) \omega (n+1),\ldots ,\omega (n+k) are mutually distinct. In this paper, we give upper and lower bounds for n k n_k . We study the same quantity when ω ( n ) \omega (n) is replaced by Ω ( n ) \Omega (n) , the total number of prime factors of n n counted with repetitions.

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .

Jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding 10 7 . Using the properties of cyclotomic polynomials, we improve this result to show that every perfect number is divisible by a prime exceeding 10 8 .

Let p be a prime number, λ be an integer. We obtain new results related to the congruence x1x2≡x3x4+λ(modp).

Here, we construct infinitely many number fields of any given degree d > 1 whose class numbers have many prime factors.

Let P be the set of all primes. The following result is proved: For any nonzero integer a ,theset a + P contains arbitrarily long sequences which have the same largest prime factor. We give an application to thedynamics of the w function which extends the “seven” in Theorem 2.14 of [Wushi Goldring, Dynamics ofthe w function and primes, J. Number Theory 119 (2006) 86–98] to any positive integer. Beyond this wealso establish a relation between a result of congruent covering systems and a question on the dynamicsof the w function. This implies that the answer to Conjecture 2.16 of Goldring’s paper is negative. Twoconjectures are posed. © 2008 Elsevier Inc. All rights reserved. MSC: 11A41; 37B99; 11N36 Keywords: Largest prime factor; Dynamics; Sequence; Parent 1. Introduction Let P be the set of all (rational) primes. For any positive integer n let P(n) denote the largestprime factor of n .Let P( 0 ) =2. Let A 3 = pqr 3 p,q,r ∈ P \ p p ∈ P. ✩ Supported by the National Natural Science Foundation of China, Grant No. 10771103.

We introduce explicit bounds for the sum , where is the number of primes that are not greater than n.

Let pn be the n-th prime number and xn = p n+1 n+1 /pnn. We show that the sequence (xn)n?N is not monotonic for any integer N > 1 and that the series +??1n=1/xn is divergent. Related series are studied as well.

Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

A positive integer is called a (Ore's) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than 1 is even. If Ore's conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than 1 must be divisible by a prime greater than 10 5 .

For an integer v > 1, we denote by co(v) and P(v) the number of distinct prime divisors of v and the greatest prime factor of v, respectively, and we put co(l) = O, P(1) = 1. Further we write zra(v) for the number of primes ~ k. Let d = 1. A well known theorem of Sylvester [15] states that

We consider the number of r-tuples of squarefree numbers in a short interval. We prove that it cannot be much bigger than the expected value and we also establish an asymptotic formula if the interval is not very short.

The original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on [0, 1] to give a Chebyshev-type lower bound in prime number theory. We study a generalization of thismethod for polynomials inmany variables. Ourmain result is a lower bound for the integral of Chebyshev's ψ-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support.

Let N be sufficiently large odd integer. It is proved that the equation N=n1+n2+n3 has solutions, where ni has a fixed number of prime factors, and an asymptotic formula holds for the number of representations.

To every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the minimum number of wild primes. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes.

In this paper we establish the distribution of prime numbers in a given arithmetic progression p ≡ l for which ap + b is squarefree.

Using a theorem on linear forms in logarithms, we show that the equation px−2y=pu−2v has no solutions (p,x,y,u,v) with x≠u, where p is a positive prime and x,y,u, and v are positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 1015. More generally, we obtain a similar result for px−qy=pu−qv>0 where q is a positive prime, q≢1 mod 12 . We solve a question of Edgar showing there is at most one solution (x,y) to px−qy=2h for positive primes p and q and positive integer h. Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions (x,y) to |px±qy|=c and at most two solutions (x,y,z) to px±qy±2z=0, for given positive primes p and q and integer c.

We study prime and composite numbers in the sequence of integer parts of powers of a fixed real number. We first prove a result which implies that there is a transcendental number ξ>1 for which the numbers [ξn !], n =2,3, ..., are all prime. Then, following an idea of Huxley who did it for cubics, we construct Pisot numbers of arbitrary degree such that all integer parts of their powers are composite. Finally, we give an example of an explicit transcendental number ζ (obtained as the limit of a certain recurrent sequence) for which the sequence [ζn], n =1,2,..., has infinitely many elements in an arbitrary integer arithmetical progression.

Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5×1010, yielding the counting function π2,4(5 × 1010) = 118905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computedB 4(5×1010)=1.197054473029 andB 4=1.197054±7×10−6. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5×1010, yielding the counting function π2,6(5 × 1010) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computedB 6(5×1010)=0.93087506039231 andB 6=1.135835±1.2×10−6.

For every positive integer n and every finite set S of prime numbers, we construct An-extensions of Q unramified at all primes in S∪{∞}; moreover, these extensions are obtained as splitting fields of totally real monic polynomials in Z [X] of degree n whose discriminant is not divisible by any prime number p in S. As a corollary, we obtain that there exist infinitely many linearly disjoint tamely ramified An-extensions of Q .

We prove an asymptotic formula for the number of representations of a sufficiently large positive integer N as the sum of two primes and the square of a natural number when they are almost equal.

In 1965 Erdős introduced f2(s): f2(s) is the smallest integer such that every l>f2(s) is the sum of s distinct primes or squares of primes where a prime and its square are not both used. We prove that for all sufficiently large s, f2(s)⩽p2+p3+⋯+ps+1+3106, and the set of s with the equality has the density 1.

Consider a sequence of l positive integers a1,a2,…,al with aj−aj−1=k(>0), j=2,3,…,l. An element b of A is called a relative prime number if b is a relatively prime to any element a of A with a≠b. In this paper, we show that for every k, there is a positive integer l0(k) such that for all integers l⩾l0(k), there exists a sequence A with length l which has no relative prime number.

Let D be a subset of the set P of prime numbers not containing any twin primes. Kemnitz and Kolberg raised the following question. For any given natural number n, are there only finitely many such minimal sets D, of the size n, such that the induced prime distance graph has chromatic number 4? In this paper, a conditional answer to this question based on a well-known conjecture from the prime number theory is given.

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.

We establish, utilizing the Hardy-Littlewood circle method, an asymptotic formula for the number of pairs of primes whose differences lie in the image of a fixed polynomial. We also include a generalization of this result where differences are replaced with any integer linear combination of two primes.

The following assertion is proved. Let gbe a q-multiplicative function for which vg(n)v = 1 for every integer, and g(p) = constantfor every large prime p. Then there is some integer kin [1; c] such that g k(nq) = 1 for every n∈ N. Here cis a suitable absolute constant.

In this paper we study the distribution modulo 1 of the sequence of vectors (pα1, …, pαk), where k⩾2 is a fixed integer and α1, …, αk are fixed real numbers lying in the interval (0, 1) and p runs over the set of prime numbers.

It is well known that a nonconstant polynomial f (x) with integer coefficients produces, for integer values of x, at least one composite image. In this note, we use Taylor expansions to improve this elementary result, showing that f (x) takes an infinite number of composite values. Given a positive integer n, we show that f (x) takes an infinite number of values that are divisible by at least n distinct primes, and an infinite number of values that are divisible by pn for some prime p.

We prove that if K is a finite extension of Q , P is the set of prime numbers in Z that remain prime in the ring R of integers of K , f , g ∈ K [ X ] with deg g >deg f and f , g are relatively prime, then f + pg is reducible in K [ X ] for at most a finite number of primes p ∈ P . We then extend this property to polynomials in more than one indeterminate. These results are related to Hilbert's irreducibility theorem.

The ℬ-free number is a generalization of the well-known squarefree numbers. The result that if θ > 33/80, then the interval [x-xθ,x] must contain a B-free number for largex has been proved.

In a paper of 1933, D. H. Lehmer continued Pierce's study of integral sequences associated to polynomials generalizing the Mersenne sequence. He developed divisibility criteria, and suggested that prime apparition in these sequences — or in closely related sequences — would be denser if the polynomials were close to cyclotomic, using a natural measure of closeness.We review briefly some of the main developments since Lehmer's paper, and report on further computational work on these sequences. In particular, we use Mossinghoff's collection of polynomials with smallest known measure to assemble evidence for the distribution of primes in these sequences predicted by standard heuristic arguments.The calculations lend weight to standard conjectures about Mersenne primes, and the use of polynomials with small measure permits much larger numbers of primes to be generated than in the Mersenne case.

This version of the paper corrects an inaccuracy in the proof of Theorem 2.9 in the published version. The main results remain unchanged., Corrected arXiv version of the published paper