Integers without large prime factors

Suppose {X(t), t ∈ G d } is a random field in d dimensions, with d ∈ ℤ+; that is, {X(t), t ∈ G d } is a collection of random variables X(t) taking values in a state space S, defined on a probability space (Ω, A, P), and indexed by the variable t ∈ G d . Throughout this chapter, G will stand for either the set of real numbers ℝ, or the set of integers ℤ; thus, the random field {X(t)} is allowed to “run” in either continuous or discrete “time.” Similarly, G+ will denote ℝ+ or ℤ+ (the sets of positive real numbers and integers, respectively) according to whether G = ℝ or G = ℤ.

In 1995, A. Wiles [1,2] announced a proof of Fermat's Last Theorem, which is stated as follows: If \(\pi\) is an odd prime and \(x\), \(y\), \(z\) are relatively prime positive integers, then \(z^\pi\not=x^\pi+y^\pi.\) In this note, a simpler proof of this theorem is offered. It is proved that if \(\pi\) is an odd prime and \(x\), \(y\), \(z\) are positive integers satisfying \(z^\pi=x^\pi+y^\pi,\) then \(x\), \(y,\) and \(z\) are each divisible by \(\pi\). The proof offered by Wiles uses cyclic groups. It is the purpose of this note to offer a proof of this result, with techniques within the grasp of Fermat.

In 1995, A, Wiles announced, using cyclic groups ( a subject area which was not available at the time of Fermat), a proof of Fermat's Last Theorem, which is stated as follows: If is an odd prime and x; y; z; are relatively prime positive integers, then z 6= x + y: In this note, an elegant proof of this result is given. It is proved, using elementary algebra, that if is an odd prime and x; y; z are positive integers satisfying z = x+y; then z; x; y are each divisible by :

The article is devoted to the problem of the real numbers' decomposition to a linear combination of natural numbers' squares. First of all the theorem giving indirectly the conditions of existing natural numbers' two equal squares is formulated and proved. Basing on this theorem two algorithms designed for finding couples of integer numbers generating eąual sums of sąuares and determining the multiplicity of that couples are presented. Finally the calculation of the algorithms' classes and estimation of the calculation errors is made. Also an example output of presented algorithms was shown.

Let be a positive integer and /(n) the Moebius function.If n > 1, let P(n) denote its largest prime factor and put P(\) = 1.We study the asymptotic behavior of the sum M*(x, y) = 1xx,n*.x,p(")<yi>.(n)as x, y - oo and discuss a few applications.(1.3) *(x,y}~ 2 1respectively.If a remains fixed as x - oo, it is known (see [5,4]) that(1.5) lim *<* *'/a> = p(") x -

On the periodic solutions emerging from the cylindrical precession of an axisymmetric satellite in a circular orbit

Structure analysis of periodically controlled chemical processes

This paper presents a k-ary Montgomery modular inverse algorithm over nonbinary computers by using Sedjelmaci's right shift k-ary greatest common divisor scheme. Over traditional binary computers, Kaliski's scheme can output Montgomery modular inverse Q − 12n mod P, where P is coprime to Q and n is the bit length of P. Over k-ary computers, our algorithm can discover the k-ary Montgomery inverse Q − 1km mod P, where P, Q, and k are pairwise relatively prime positive integers and m = log kP. In the worst case, the computational cost of our algorithm is O(m2)k-ary digit operations. Copyright © 2013 John Wiley & Sons, Ltd.

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of ζ(1/2 + it) is bounded by \(\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)}\) , we show that for any given α > 0 the interval \((X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ]\) contains an integer having no prime factor exceeding Xα for all X sufficiently large.

Duality between prime factors and an application to the prime number theorem for arithmetic progressions

We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ε, we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding nε, with the length of the strings tending to infinity with speed log log log log n.

For any fixed integer a and real variables x , y with y x let N a ( x, y ) denote the number of primes p ≤ x for which p + a has at least one prime factor greater than y . As an elementary application of the following deep theorem of Bombieri on arithmetic progressions, Theorem (Bombieri, [1]). For any constant A > 0, there exists a positive constant B such that if with l = log x, then for x > 1 where Φ(x; m, a) denotes the number of primes less than x which are congruent to a mod m ; we shall prove the following theorem: Theorem 1. Let a be any fixed integer and let x > e. We then have where the double sum is taken over primes p and q .

Let λ> 0, δ ∈]0,1[ and f(n) be a multiplicative function satisfying essentially for a large x ; � f (p)≤x log f(p )= λx+ O(x(log x) −δ ). In the present work, we estab

A fundamental result, due to Sabidussi and Vizing, states that every connected graph has a unique prime factorization rel- ative to the Cartesian product; but disconnected graphs are not uniquely prime factorable. This paper describes a system of modular arithmetic on graphs under which both connected and disconnected graphs have unique prime Cartesian factorizations. The Cartesian product of two simple graphs G = (V (G); E(G)) and H = (V (H); E(H)) is the graph G H with V (G H) = V (G) V (H), and (u; x)(v; y) 2 E(G H) if either u = v and xy 2 E(H), or uv 2 E(G) and x = y. This product is commutative and associative: G H = H G and G (H K) = (G H) K (up to isomorphism) for all graphs G, H and K. Also G H is connected if and only if both G and H are connected. For a full treatment of this product, see Chapter 4 of Imrich and Kla zar (2). We denote the empty graph (i.e. the graph with no vertices) as O, and the complete graph on n vertices as Kn. Notice that G O = O and G K1 = G for all graphs G. If n 2 N, then nG denotes the graph that is the disjoint union of n copies of G (or O if n = 0). Note n(G H) = nG H = G nH. For a positive integer n, we dene G n = G G G (n factors) and we adopt the convention G 0 = K1. A graph G is prime if it is nontrivial and G = G1 G2 implies G1 = K1 or G2 = K1. Every graph G has a prime factorization G = G1 G2 Gp, where each factor Gi is prime. A fundamental theorem, proved indepen- dently by Sabidussi (3) and Vizing (4) states that the prime factorization of a connected graph is unique, that is if a connected graph G has prime factor- izations G1 G2 Gp and H1 H2 Hq, then p = q and Gi = Hi for 1 i p (after reindexing, if necessary). But disconnected graphs are not uniquely prime factorable, in general. One standard example is the graph G = K1 + K2 + K 2

We discuss the procedure of different partitions in the finite set of N integer numbers and construct generic formulas for a bijective map of real numbers sy, where y = 1, 2,…, N, N = \( \underset{k=1}{\overset{n}{\varPi}}{X}_k, \) and Xk are positive integers, onto the set of numbers s(y(x1, x2,…, xn)). We give the functions used to present the bijective map, namely, y(x1, x2, …, xn) and xk(y) in an explicit form and call them the functions detecting the hidden correlations in the system. The idea to introduce and employ the notion of “hidden gates” for a single qudit is proposed. We obtain the entropic-information inequalities for an arbitrary finite set of real numbers and consider the inequalities for arbitrary Clebsch–Gordan coefficients as an example of the found relations for real numbers.