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This paper aims at presenting pairs of rectangles representing the same sphenic number where, in each pair, the sum of the areas is 2 times sphenic number -1

In this paper, we study the symmetry between the supersingular prime number according to smallest sphenic number. With this symmetry, we show that the elements of sporadic group generates all prime numbers with the order by a simple application.

We consider for integers [Formula: see text] the [Formula: see text]-generalized Fibonacci sequences [Formula: see text], whose first [Formula: see text] terms are [Formula: see text] and each term afterwards is the sum of the preceding [Formula: see text] terms. We give a lower bound for the largest prime factor of the sum of two terms in [Formula: see text]. As a consequence of our main result, for every fixed finite set of primes [Formula: see text], there are only finitely many positive integers [Formula: see text] and [Formula: see text]-integers which are a non-trivial sum of two [Formula: see text]-Fibonacci numbers, and all these are effectively computable.

In this paper, we consider the representation of a large positive integer N ≡ 4 ( mod 24 ) in the form p 2 + x 1 2 + x 2 2 + x 3 2 where p is a prime number and x 1 , x 2 , x 3 are almost-primes. A positive integer is called a P r -number if its number of prime factors counted according to multiplicity is at most r. We establish the above representation in the following two different forms. (i) x 1 is a P 2 -number and x 2 x 3 is a P 182 -number; (ii) x 1 x 2 x 3 is a P 12 -number.

Text A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors p 1 , … , p t of n and positive integers α 1 , … , α t such that n = p 1 α 1 + ⋯ + p t α t . It is clear that t ≥ 3 . In 1992, Erdős and Hegyvari proved that, for any prime p, there exist infinitely many weakly prime-additive numbers with t = 3 which are divisible by p. In this paper, we prove that, for any positive integer m, there exist infinitely many weakly prime-additive numbers with t = 3 which are divisible by m if and only if 8 ∤ m . We also present some related results and pose several problems for further research. Video For a video summary of this paper, please visit https://youtu.be/WC_VRFtY07c .

In this paper, it is proved by a different method that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 17 powers of 2, which improves the previous result $$k=31$$ .

Let p be an odd prime and let m be a positive p -th power free integer greater than one. Let K be an algebraic number field generated by the positive p -th root of m over the field of rational numbers, L its Galois closure in the field of complex numbers, and k the p -th cyclotomic field. We calculate a class number relation among them, which gives a sufficient condition for the class number of K to be a multiple of p . Moreover, we show that a rational prime q unequal to p is a divisor of the class number of L if and only if q divides that of k or K .

For the specific set of fifteen ternary quadratic forms x 2 + b y 2 + c z 2 , b , c ∈ { 1 , 2 , 4 , 8 } , ( b , c ) ∈ { ( 2 , 16 ) , ( 8 , 16 ) , ( 1 , 3 ) , ( 2 , 3 ) , ( 1 , 5 ) } , it is shown that the distinct zero-free representations of an odd prime by these forms depend upon the class numbers h ( − k p ) , k ∈ { 1 , 3 , 4 , 5 , 8 , 12 , 20 , 24 } . We determine when such a form is universal zero-free for an arithmetic progression of primes, i.e., when a prime from such a progression can be represented without zero components. The exceptional primes, which cannot be represented in this way, fall into two distinct classes. They are either infinite in number and belong to arithmetic progressions of primes, so-called infinite exceptional sets, or they are finite in number and build so-called finite exceptional sets. These exceptional sets are determined. Moreover, we show how to derive the finite number of primes expressible by a form x 2 + b y 2 + c z 2 in essentially m ways, and illustrate the method. Reinterpreting results by Dirichlet [12] , Dickson [10] and Kaplansky [23] , we show that the forms ( b , c ) ∈ { ( 1 , 2 ) , ( 1 , 3 ) , ( 2 , 3 ) , ( 2 , 4 ) } are the only strictly universal zero-free forms of type x 2 + b y 2 + c z 2 , i.e., they can be represented without zero components for all primes up to a known finite number of primes.

This paper again specifies the major points of the article “Do Prime Numbers Obey a Three-Dimensional Double Helix?” [1] which was received on February 16, 2006 by Hadronic Journal. New information has been added and elucidated upon, such as why the numbers 2 and 3 are not considered true prime numbers, and why s in the following formulas for 6s - 1 and for 6s + 1 is really a composite number equal to the sum of two other numbers, suggesting that s is always to be considered as an integer. Other new information is added as well, such as how an engineer in a matter of seconds decomposed a large prime product into its constituent primes using basic software and won a contract for his firm.

For an odd prime number p, let h−p denote the relative class number of the pth cyclotomic field . It is conjectured that h−p is odd when p is of the form p = 2l + 1 with an odd prime number l, and ...

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramer on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun–Titchmarsh theorem.

Let N0 be the set of natural numbers whose binary expansions have an even number of 1’s, and let N1 = N\N0. In this paper, we obtain asymptotic formulas for the number of primes p not exceeding X and such that p ∈ N i , p + 1 ∈ N j , where i and j take values 0 and 1 independently of each other.

We consider the distribution of the largest prime divisor of the integers in the interval [2, x] and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving an asymptotic formula for this mode as x tends to infinity, we look at the set of those prime numbers which, for some value of x, occur most frequently as the largest prime divisor of the integers in the interval [2, x]. We find that many prime numbers never have this property. We compare the set of “popular primes,” those primes which are at some point the mode, to other interesting subsets of the prime numbers. Finally, we apply the techniques developed to a similar problem which arises in the analysis of factoring algorithms.

In this article the why and how of the prime numbers were shown; to be more specific, I present the pattern that was defined, i.e. every prime number is in the interval between (30a + (p)) and (42a + (p1)); p = (11;17;23;29); p1 = (13; 19; 25; 31; 37; 43). This verifies the accuracy of the series of Dirichelet, and improvement, because any series that of a prime number matches the prime number of this pattern. Another contribution of this work is to know whether a number is prime; both for a small number, as for one he is infinitely large, without applying the process of factorization.

Let K be a positive integer, { A i , 1 ≤ i ≤ K } be any partition of the sequence of squares of primes and s ( K ) be the smallest positive integer such that every sufficiently large integer can be written as the sum of no more than s ( K ) elements, which belong to one of the sets A i . In this paper, we prove that s ( K ) ≪ ϵ K 2 + ϵ for sufficiently small positive number ϵ and all K ≥ 1 .

Let q ≥ 2 be an integer and S q ( n ) denote the sum of the digits in base q of the positive integer n . It is proved that for every real number α and β with α β , lim x ⟶ + ∞ ⁡ 1 x ♯ { n ≤ x : α ≤ v ( φ ( n ) ) − 1 2 b ( log ⁡ log ⁡ n ) 2 1 3 b ( log ⁡ log ⁡ n ) 3 2 ≤ β } = 1 2 π ∫ α β e − t 2 2 d t , where v ( n ) is either ω ˜ ( n ) or Ω ˜ ( n ) , the number of distinct prime factors and the total number of prime factors p of a positive integer n such that S q ( p ) ≡ a mod b ( a , b ∈ Z , b ≥ 2 ). This extends the results known through the work of P. Erdős and C. Pomerance, M.R. Murty and V.K. Murty to primes under digital constraint.

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 .

We study the properties of prime number sequences obtained using a well-defined equivalence relation . It will be seen that the elements of each class of are all prime numbers which constitute the fundamental object of our study. The number of prime numbers of each class less than or equal to a given quantity , the number of the different equivalence classes and some other results will be deduced.

We prove that every integer greater than two may be written as the sum of a prime and a square-free number.

For a positive integer n, let σ(n) and γ(n) denote the sum of divisors and the product of distinct prime divisors of n, respectively. It is known that, if σ(n)=γ(n)2, then at most two exponents of odd primes are equal to 1 in the prime factorization of n. In this paper, we prove that, if σ(n)=γ(n)2 and only one exponent is equal to 1 in the prime factorization of n, then (1) n is divisible by 3; (2) n is divisible by the fourth powers of at least two odd primes; (3) at least two exponents of odd primes are equal to 2. We also prove that, if σ(n)=γ(n)2, then at least half of the exponents α of the primes have the property that the numbers α+1 must be either primes or prime squares.

It is known, that any prime number has presentation in linear and quadratic forms. These properties may be used for finding class subsets of prime numbers. For that it is showed, that all prime numbers simple quadratic forms consist of a2+mb2, m=1,±2,3 . On these grounds it is examination for variants of prime numbers classification. It is discovered eight non-intersecting subsets of prime numbers, in conformity with equivalence classes modulo 24. The proposed classification is used for analyses Mersenne and Fermat numbers and composite numbers.

A prime number p is called strong if p − 1 and p + 1 each have a large prime factor. Let x be a large positive number. We show a lower bound on the number of strong primes p between 1 and x such that p−1 and p+1 each have a prime factor larger than \({\sqrt{p}}\) .

In a series of papers, we constructed large families of normal numbers using the distribution of the values of the largest prime factor function. Here, letting $$p(n)$$ stand for the smallest prime factor of $$n$$ , we show how a concatenation of the successive values of $$p(n)$$ can yield a normal number in any given basis $$q\ge 2$$ . We further expand on this idea to create various large families of normal numbers.

Binary decomposition of numbers forms geometrical blocks. They depend on approach of a prime or whole prime number to values of the two in the degree equal to number from a natural row. As a result there is a strict geometry among prime or whole prime numbers in the form of block structure. This structure receives distinctive signs and harmonicas in positive part of a number of the whole prime numbers are shown. Statistically from a natural numbers regularities of growth of power of the left and right reference points, as borders previous and the subsequent from values of the two in degree, at blocks increasing on length among the whole prime numbers are proved.

Statistical modeling by asymmetric waves, with variables amplitude and a half-cycle of fluctuation, dynamics of a scatter of block structure of positive part of a number of the integers prime which located in a row of 10 million natural numbers, proved emergence of three stages of growth of the left and right reference points in blocks of binary decomposition of prime numbers. These a reference point settle down on each side from the dividing line in the form of the two in the degree equal to number of the category of a binary numeral system, without unit. The first stage of critical chaos is formed by critical prime numbers 0, 1 and 2. The second stage of an accruing order begins with number 3 and comes to the end with a margin error in 1% at the 1135th category of binary notation for the left reference point. At blocks increasing on length among the integers prime by calculations after the 1135th category there comes the third stage with high definiteness of the beginning and the end of blocks of binary decomposition of positive prime numbers.

The integer split quaternions form a noncommutative algebra over ℤ. We describe the prime and maximal spectrum of the integer split quaternions and investigate integer-valued polynomials over this ring. We prove that the set of such polynomials forms a ring, and proceed to study its prime and maximal ideals. In particular we completely classify the primes above 0, we obtain partial characterizations of primes above odd prime integers, and we give sufficient conditions for building maximal ideals above 2.

It is proved that every sufficiently large odd integer is the sum of one prime, two squares of primes and 35 powers of 2. This improves a previous result with 35 replaced by 83.

Let R ( n ) denote the number of representations of the positive number n as the sum of one prime and two squares of primes. It is established that the anticipated asymptotic formula for R ( n ) fails for at most O ( N 1 / 3 + e ) positive integers not exceeding N.

International audience; Let ω(n) and Ω(n) denote, respectively, the total number of prime factors and the number of distinct prime factors of the integer n. Euler proved that an odd perfect number N is of the form N = pᶱm² where p ≡ e ≡ 1 (mod 4), p is prime, and p ∤ m. This implies that Ω(N) ≥ 2ω(N) − 1. We. We prove that Ω(N) ≥ (18ω(N) −31) /7andΩ(N) ≥ 2ω(N) + 51.

The main objective of this paper is to develop an infinite product formula for the ratio of consecutive prime numbers, using Jacobi elliptic functions.

Let p be a prime number, and let 𝔹p,n be the nth layer in the cyclotomic ℤp-extension of ℚ with ring of integers [Formula: see text]. In this paper we show that for each even integer m ≥ 2 and each prime number p the orders of the Quillen K-groups [Formula: see text] are unbounded, and that there are in fact infinitely many different prime numbers dividing the order of [Formula: see text] for some n.

We study the asymptotic behavior of the sequence with general term consisting of the ratio An by Gn, the arithmetic and geometric means of the prime numbers p1, p2, …, pn, respectively, in which, pn denotes the nth prime number.

The present algebraic development begins by an exposition of the data of the problem. The definition of the primal radius r>0 is : For all positive integer x≥3 exists a finite number of integers called the primal radius r>0, for which x+r and x-r are prime numbers. The corollary is that 2x=(x+r)+(x-r) is always the sum of a finite number of primes. Also, for all positive integer x≥0, exists an infinity of integers r>0, for which x+r and r-x are prime numbers. The conclusion is that 2x=(x+r)-(r-x) is always an infinity of differences of primes.

We show that for any integers a and m with m ≥ 1 and gcd(a,m) = 1, there is a solution to the congruence pr ≡ a (modm) where p is prime, r is a product of at most k = 17 prime factors and p, r ≤ m. This is a relaxed version of the still open question, studied by P. Erdős, A. M. Odlyzko and A. Sarkozy, that corresponds to k = 1 (that is, to products of two primes).

Recently, Pollack and Shevelev [‘On perfect and near-perfect numbers’, J. Number Theory 132 (2012), 3037–3046] introduced the concept of near-perfect numbers. A positive integer $n$ is called near-perfect if it is the sum of all but one of its proper divisors. In this paper, we determine all near-perfect numbers with two distinct prime factors.

Estimates are obtained for the number of natural numbers below a parameter that do not have a representation as the sum of two squares of primes and a kth power of a prime. These improve earlier bounds in the order of magnitude. The method is then also applied to some related questions.

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.

Recently we discussed in a letter the modular arithmetic and a particular integer numbers (odd) model, which we then utilized to discuss parsed integer numbers statistics and nearness. We continue in this letter to discuss the approach as applied to other models, namely the triangular embedding of symmetric copies in large number representations, and obtain a relationship for identifying primes from non primes. We then discuss other models that may prove useful such as the golden ratio spiral and number sequences from the same approach.

A number is cubefull if the exponent of every prime in its prime factorization is at least three. In this paper, we give, for a fixed l, the number of pairs of cubefull numbers n, n + l such that n is less than a given quantity.

The asymptotic order of the number of primes, which are such that the shift by a fixed integer is a number supported by a given set of primes times a coprime squarefree number, is determined. The order is also determined when the shift and its negative have this same shape. In each case the order is dependent only on the set of primes and the squarefree core of the shift.

It is proved that an integer n?=?2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial fwith coefficients in Zn, the ring of integers modulo n, such that each element of Znis a root of f. This classroom note could find use in any introductory course on abstract algebra or elementary number theory.

A cyclotomic polynomial \Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family \Phi_{pqr}(x) with p, Comment: To appear in Involve, 23 pages, 7 Tables, Question 5 has been meanwhile answered in the positive by Eugenia Rosu; (extended version, 32 pages)

It is proved that every sufficiently large even integer is a sum of one prime, one square of prime, two cubes of primes and 161 powers of 2.

It is known that innitely many number elds and function m over Fq(T ) with class number indivisible by an arbitrary prime '. We give an explicit description of those primes (and prime powers) q for which the result holds. For the special case where ' = 3 and m = 2, we recover Ichimura's result.