The Math Encyclopedia of Smarandache Type Notions [Vol. I
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Further Generalizations of Happy Numbers
Further Generalizations of Happy Numbers E. S. Williams June 15, 2016 Abstract In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple in [2] and establish further results, not given in that work. Then we construct a similar function expanding the definition of happy numbers to negative integers. Using this function, we compute and prove results extending those regarding higher powers and sequences of consecutive happy numbers that El-Sidy and Siksek and Grundman and Teeple proved in [1], [2] and [3] to negative integers. Finally, we consider a variety of special cases, in which the existence of certain fixed points and cycles of infinite families of generalized happy functions can be proven. 1 Introduction Happy numbers have been studied for many years, although the origin of the concept is unclear. Consider the sum of the square of the digits of an arbitrary positive integer. If repeating the process of taking the sums of the squares of the digits of an integer eventually gets us 1, then that integer is happy. This paper first provides a brief overview of traditional happy numbers, then describes existing work on generalizations of the concept to other bases and higher powers. We extend some earlier results for the cubic case to the quartic case. We then extend the happy function S to negative integers, constructing a function Q : Z ! Z which agrees with S over Z+ for this purpose. We then generalize Q to various bases and higher powers. As is traditional in 1 the study of special numbers, we consider consecutive sequences of happy numbers, and generalize this study to Q. -
Generalizations of Euler Numbers and Polynomials 1
GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS QIU-MING LUO AND FENG QI Abstract. In this paper, the concepts of Euler numbers and Euler polyno- mials are generalized, and some basic properties are investigated. 1. Introduction It is well-known that the Euler numbers and polynomials can be defined by the following definitions. Definition 1.1 ([1]). The Euler numbers Ek are defined by the following expansion t ∞ 2e X Ek = tk, |t| ≤ π. (1.1) e2t + 1 k! k=0 In [4, p. 5], the Euler numbers is defined by t/2 ∞ n 2n 2e t X (−1) En t = sech = , |t| ≤ π. (1.2) et + 1 2 (2n)! 2 n=0 Definition 1.2 ([1, 4]). The Euler polynomials Ek(x) for x ∈ R are defined by xt ∞ 2e X Ek(x) = tk, |t| ≤ π. (1.3) et + 1 k! k=0 It can also be shown that the polynomials Ei(t), i ∈ N, are uniquely determined by the following two properties 0 Ei(t) = iEi−1(t),E0(t) = 1; (1.4) i Ei(t + 1) + Ei(t) = 2t . (1.5) 2000 Mathematics Subject Classification. 11B68. Key words and phrases. Euler numbers, Euler polynomials, generalization. The authors were supported in part by NNSF (#10001016) of China, SF for the Prominent Youth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, China. This paper was typeset using AMS-LATEX. 1 2 Q.-M. LUO AND F. QI Euler polynomials are related to the Bernoulli numbers. For information about Bernoulli numbers and polynomials, please refer to [1, 2, 3, 4]. -
POWER FIBONACCI SEQUENCES 1. Introduction Let G Be a Bi-Infinite Integer Sequence Satisfying the Recurrence Relation G If
POWER FIBONACCI SEQUENCES JOSHUA IDE AND MARC S. RENAULT Abstract. We examine integer sequences G satisfying the Fibonacci recurrence relation 2 3 Gn = Gn−1 + Gn−2 that also have the property that G ≡ 1; a; a ; a ;::: (mod m) for some modulus m. We determine those moduli m for which these power Fibonacci sequences exist and the number of such sequences for a given m. We also provide formulas for the periods of these sequences, based on the period of the Fibonacci sequence F modulo m. Finally, we establish certain sequence/subsequence relationships between power Fibonacci sequences. 1. Introduction Let G be a bi-infinite integer sequence satisfying the recurrence relation Gn = Gn−1 +Gn−2. If G ≡ 1; a; a2; a3;::: (mod m) for some modulus m, then we will call G a power Fibonacci sequence modulo m. Example 1.1. Modulo m = 11, there are two power Fibonacci sequences: 1; 8; 9; 6; 4; 10; 3; 2; 5; 7; 1; 8 ::: and 1; 4; 5; 9; 3; 1; 4;::: Curiously, the second is a subsequence of the first. For modulo 5 there is only one such sequence (1; 3; 4; 2; 1; 3; :::), for modulo 10 there are no such sequences, and for modulo 209 there are four of these sequences. In the next section, Theorem 2.1 will demonstrate that 209 = 11·19 is the smallest modulus with more than two power Fibonacci sequences. In this note we will determine those moduli for which power Fibonacci sequences exist, and how many power Fibonacci sequences there are for a given modulus. -
On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function
mathematics Article On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function ŠtˇepánHubálovský 1,* and Eva Trojovská 2 1 Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic 2 Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +420-49-333-2704 Received: 3 October 2020; Accepted: 14 October 2020; Published: 16 October 2020 Abstract: Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk ≡ 0 (mod n). In this paper, we shall find all positive solutions of the Diophantine equation z(j(n)) = n, where j is the Euler totient function. Keywords: Fibonacci numbers; order of appearance; Euler totient function; fixed points; Diophantine equations MSC: 11B39; 11DXX 1. Introduction Let (Fn)n≥0 be the sequence of Fibonacci numbers which is defined by 2nd order recurrence Fn+2 = Fn+1 + Fn, with initial conditions Fi = i, for i 2 f0, 1g. These numbers (together with the sequence of prime numbers) form a very important sequence in mathematics (mainly because its unexpectedly and often appearance in many branches of mathematics as well as in another disciplines). We refer the reader to [1–3] and their very extensive bibliography. We recall that an arithmetic function is any function f : Z>0 ! C (i.e., a complex-valued function which is defined for all positive integer). -
Prime Divisors in the Rationality Condition for Odd Perfect Numbers
Aid#59330/Preprints/2019-09-10/www.mathjobs.org RFSC 04-01 Revised The Prime Divisors in the Rationality Condition for Odd Perfect Numbers Simon Davis Research Foundation of Southern California 8861 Villa La Jolla Drive #13595 La Jolla, CA 92037 Abstract. It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k + 4m+1 ℓ 2αi 1) i=1 qi to establish that there do not exist any odd integers with equality (4k+1)4m+2−1 between σ(N) and 2N. The existence of distinct prime divisors in the repunits 4k , 2α +1 Q q i −1 i , i = 1,...,ℓ, in σ(N) follows from a theorem on the primitive divisors of the Lucas qi−1 sequences and the square root of the product of 2(4k + 1), and the sequence of repunits will not be rational unless the primes are matched. Minimization of the number of prime divisors in σ(N) yields an infinite set of repunits of increasing magnitude or prime equations with no integer solutions. MSC: 11D61, 11K65 Keywords: prime divisors, rationality condition 1. Introduction While even perfect numbers were known to be given by 2p−1(2p − 1), for 2p − 1 prime, the universality of this result led to the the problem of characterizing any other possible types of perfect numbers. It was suggested initially by Descartes that it was not likely that odd integers could be perfect numbers [13]. After the work of de Bessy [3], Euler proved σ(N) that the condition = 2, where σ(N) = d|N d is the sum-of-divisors function, N d integer 4m+1 2α1 2αℓ restricted odd integers to have the form (4kP+ 1) q1 ...qℓ , with 4k + 1, q1,...,qℓ prime [18], and further, that there might exist no set of prime bases such that the perfect number condition was satisfied. -
Power Values of Divisor Sums Author(S): Frits Beukers, Florian Luca, Frans Oort Reviewed Work(S): Source: the American Mathematical Monthly, Vol
Power Values of Divisor Sums Author(s): Frits Beukers, Florian Luca, Frans Oort Reviewed work(s): Source: The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.119.05.373 . Accessed: 15/02/2013 04:05 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded on Fri, 15 Feb 2013 04:05:44 AM All use subject to JSTOR Terms and Conditions Power Values of Divisor Sums Frits Beukers, Florian Luca, and Frans Oort Abstract. We consider positive integers whose sum of divisors is a perfect power. This prob- lem had already caught the interest of mathematicians from the 17th century like Fermat, Wallis, and Frenicle. In this article we study this problem and some variations. We also give an example of a cube, larger than one, whose sum of divisors is again a cube. 1. INTRODUCTION. Recently, one of the current authors gave a mathematics course for an audience with a general background and age over 50. -
An Identity for Generalized Bernoulli Polynomials
1 2 Journal of Integer Sequences, Vol. 23 (2020), 3 Article 20.11.2 47 6 23 11 An Identity for Generalized Bernoulli Polynomials Redha Chellal1 and Farid Bencherif LA3C, Faculty of Mathematics USTHB Algiers Algeria [email protected] [email protected] [email protected] Mohamed Mehbali Centre for Research Informed Teaching London South Bank University London United Kingdom [email protected] Abstract Recognizing the great importance of Bernoulli numbers and Bernoulli polynomials in various branches of mathematics, the present paper develops two results dealing with these objects. The first one proposes an identity for the generalized Bernoulli poly- nomials, which leads to further generalizations for several relations involving classical Bernoulli numbers and Bernoulli polynomials. In particular, it generalizes a recent identity suggested by Gessel. The second result allows the deduction of similar identi- ties for Fibonacci, Lucas, and Chebyshev polynomials, as well as for generalized Euler polynomials, Genocchi polynomials, and generalized numbers of Stirling. 1Corresponding author. 1 1 Introduction Let N and C denote, respectively, the set of positive integers and the set of complex numbers. (α) In his book, Roman [41, p. 93] defined generalized Bernoulli polynomials Bn (x) as follows: for all n ∈ N and α ∈ C, we have ∞ tn t α B(α)(x) = etx. (1) n n! et − 1 Xn=0 The Bernoulli numbers Bn, classical Bernoulli polynomials Bn(x), and generalized Bernoulli (α) numbers Bn are, respectively, defined by (1) (α) (α) Bn = Bn(0), Bn(x)= Bn (x), and Bn = Bn (0). (2) The Bernoulli numbers and the Bernoulli polynomials play a fundamental role in various branches of mathematics, such as combinatorics, number theory, mathematical analysis, and topology. -
A Simple Proof of the Mean Fourth Power Estimate
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze K. RAMACHANDRA A simple proof of the mean fourth power estimate for 1 1 z(2 + it) and L(2 + it;c) Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 1, no 1-2 (1974), p. 81-97 <http://www.numdam.org/item?id=ASNSP_1974_4_1_1-2_81_0> © Scuola Normale Superiore, Pisa, 1974, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A Simple Proof of the Mean Fourth Power Estimate for 03B6(1/2 + it) and L(1/2 + it, ~). K. RAMACHANDRA (*) To the memory of Professor ALBERT EDWARD INGHAM 1. - Introduction. The main object of this paper is to prove the following four well-known theorems by a simple method. THEOREM 1. and then THEOREM 2. If T ~ 3, R>2 and Ti and also 1, then THEOREM 3. Let y be a character mod q (q fixed), T ~ 3, - tx,2 ... T, (.Rx > 2), and tx.i+,- 1. If with each y we associate such points then, where * denotes the sum over primitive character mod q. -
Counting Restricted Partitions of Integers Into Fractions: Symmetry and Modes of the Generating Function and a Connection to Ω(T)
Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to !(t) Z. Hoelscher and E. Palsson ∗ December 1, 2020 Abstract - Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to h, then the nonzero terms of the generating function are unimodal, with the integer h having the most partitions. Such properties can be applied to a particular class of nonlinear Diophantine equations. We also examine partitions with even numerators. We prove that there are 2!(t) − 2 partitions of an integer t into fractions with the first x consecutive even integers for numerators and equal denominators of y, where 0 < y < x < t. We then use this to produce corollaries such as a Dirichlet series identity and an extension of the prime omega function to the complex plane, though this extension is not analytic everywhere. Keywords : Integer partitions; Restricted partitions; Partitions into fractions Mathematics Subject Classification (2020) : 05A17 1 Introduction Integer partitions are a classic part of number thoery. In the most general, unrestricted case, one seeks to express positive integers as the sum of smaller positive integers. Often the function p(n) is used to denote the count of the partitions of n. No simple formula for this is known, though a generating function can be written [1]. -
A Member of the Order of Minims, Was a Philosopher, Mathematician, and Scientist Who Wrote on Music, Mechanics, and Optics
ON THE FORM OF MERSENNE NUMBERS PREDRAG TERZICH Abstract. We present a theorem concerning the form of Mersenne p numbers , Mp = 2 − 1 , where p > 3 . We also discuss a closed form expression which links prime numbers and natural logarithms . 1. Introduction. The friar Marin Mersenne (1588 − 1648), a member of the order of Minims, was a philosopher, mathematician, and scientist who wrote on music, mechanics, and optics . A Mersenne number is a number of p the form Mp = 2 − 1 where p is an integer , however many authors prefer to define a Mersenne number as a number of the above form but with p restricted to prime values . In this paper we shall use the second definition . Because the tests for Mersenne primes are relatively simple, the record largest known prime number has almost always been a Mersenne prime . 2. A theorem and closed form expression Theorem 2.1. Every Mersenne number Mp , (p > 3) is a number of the form 6 · k · p + 1 for some odd possitive integer k . Proof. In order to prove this we shall use following theorems : Theorem 2.2. The simultaneous congruences n ≡ n1 (mod m1) , n ≡ n2 (mod m2), .... ,n ≡ nk (mod mk) are only solvable when ni = nj (mod gcd(mi; mj)) , for all i and j . i 6= j . The solution is unique modulo lcm(m1; m2; ··· ; mk) . Proof. For proof see [1] Theorem 2.3. If p is a prime number, then for any integer a : ap ≡ a (mod p) . Date: January 4 , 2012. 1 2 PREDRAG TERZICH Proof. -
The Pseudoprimes to 25 • 109
MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 151 JULY 1980, PAGES 1003-1026 The Pseudoprimes to 25 • 109 By Carl Pomerance, J. L. Selfridge and Samuel S. Wagstaff, Jr. Abstract. The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Car- michael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences. 1. Introduction. According to Fermat's "Little Theorem", if p is prime and (a, p) = 1, then ap~1 = 1 (mod p). This theorem provides a "test" for primality which is very often correct: Given a large odd integer p, choose some a satisfying 1 <a <p - 1 and compute ap~1 (mod p). If ap~1 pi (mod p), then p is certainly composite. If ap~l = 1 (mod p), then p is probably prime. Odd composite numbers n for which (1) a"_1 = l (mod«) are called pseudoprimes to base a (psp(a)). (For simplicity, a can be any positive in- teger in this definition. We could let a be negative with little additional work. In the last 15 years, some authors have used pseudoprime (base a) to mean any number n > 1 satisfying (1), whether composite or prime.) It is well known that for each base a, there are infinitely many pseudoprimes to base a. -
Arxiv:Math/0201082V1
2000]11A25, 13J05 THE RING OF ARITHMETICAL FUNCTIONS WITH UNITARY CONVOLUTION: DIVISORIAL AND TOPOLOGICAL PROPERTIES. JAN SNELLMAN Abstract. We study (A, +, ⊕), the ring of arithmetical functions with unitary convolution, giving an isomorphism between (A, +, ⊕) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell- Everett[4] between the ring (A, +, ·) of arithmetical functions with Dirichlet convolution and the power series ring C[[x1,x2,x3,... ]] on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units. 1. Introduction The ring of arithmetical functions with Dirichlet convolution, which we’ll denote by (A, +, ·), is the set of all functions N+ → C, where N+ denotes the positive integers. It is given the structure of a commutative C-algebra by component-wise addition and multiplication by scalars, and by the Dirichlet convolution f · g(k)= f(r)g(k/r). (1) Xr|k Then, the multiplicative unit is the function e1 with e1(1) = 1 and e1(k) = 0 for k> 1, and the additive unit is the zero function 0. Cashwell-Everett [4] showed that (A, +, ·) is a UFD using the isomorphism (A, +, ·) ≃ C[[x1, x2, x3,... ]], (2) where each xi corresponds to the function which is 1 on the i’th prime number, and 0 otherwise. Schwab and Silberberg [9] topologised (A, +, ·) by means of the norm 1 arXiv:math/0201082v1 [math.AC] 10 Jan 2002 |f| = (3) min { k f(k) 6=0 } They noted that this norm is an ultra-metric, and that ((A, +, ·), |·|) is a valued ring, i.e.