Problem Solving and Recreational Mathematics
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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]. -
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). -
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. -
Balanced Ternary Arithmetic on the Abacus
Balanced Ternary Arithmetic on the Abacus What is balanced ternary? It is a ternary number system where the digits of a number are powers of three rather than powers of ten, but instead of adding multiples of one or two times various powers of three to make up the desired number, in balanced ternary some powers of three are added and others are subtracted. Each power of three comprisising the number is represented by either a +, a -, or 0, to show that the power of three is either added, subtracted, or not present in the number. There are no multiples of a power of three greater than +1 or -1, which greatly simplifies multiplication and division. Another advantage of balanced ternary is that all integers can be represented, both positive and negative, without the need for a separate symbol to indicate plus or minus. The most significant ternary digit (trit) of any positive balanced ternary number is + and the most significant trit of any negative number is -. Here is a list of a few numbers in decimal, ternary, and balanced ternary: Decimal Ternary Balanced ternary ------------------------------------------------- -6 - 20 - + 0 (-9+3) -5 - 12 - + + (-9+3+1) -4 - 11 - - ( -3-1) -3 - 10 - 0 etc. -2 - 2 - + -1 - 1 - 0 0 0 1 1 + 2 2 + - 3 10 + 0 4 11 + + 5 12 + - - 6 20 + - 0 Notice that the negative of a balanced ternary number is formed simply by inverting all the + and - signs in the number. Thomas Fowler in England, about 1840, built a balanced ternary calculating machine capable of multiplying and dividing balanced ternary numbers. -
Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube. -
Labeled Factorization of Integers
Labeled Factorization of Integers Augustine O. Munagi John Knopfmacher Centre for Applicable Analysis and Number Theory School of Mathematics, University of the Witwatersrand Wits 2050, Johannesburg, South Africa [email protected] Submitted: Jan 5, 2009; Accepted: Apr 16, 2009; Published: Apr 22, 2009 Mathematics Subject Classification: 11Y05, 05A05, 11B73, 11B13 Abstract The labeled factorizations of a positive integer n are obtained as a completion of the set of ordered factorizations of n. This follows a new technique for generating ordered factorizations found by extending a method for unordered factorizations that relies on partitioning the multiset of prime factors of n. Our results include explicit enumeration formulas and some combinatorial identities. It is proved that labeled factorizations of n are equinumerous with the systems of complementing subsets of {0, 1,...,n − 1}. We also give a new combinatorial interpretation of a class of generalized Stirling numbers. 1 Ordered and labeled factorization An ordered factorization of a positive integer n is a representation of n as an ordered product of integers, each factor greater than 1. The set of ordered factorizations of n will be denoted by F (n), and |F (n)| = f(n). For example, F (6) = {6, 2.3, 3.2}. So f(6) = 3. Every integer n> 1 has a canonical factorization into prime numbers p1,p2,..., namely m1 m2 mr n = p1 p2 ...pr , p1 <p2 < ··· <pr, mi > 0, 1 ≤ i ≤ r. (1) The enumeration function f(n) does not depend on the size of n but on the exponents mi. In particular we define Ω(n)= m1 + m2 + ··· + mr, Ω(1) = 0. -
A Subspace Theorem for Manifolds
A subspace theorem for manifolds Emmanuel Breuillard and Nicolas de Saxcé February 5, 2021 Abstract We prove a theorem that generalizes Schmidt’s Subspace Theorem in the context of metric diophantine approximation. To do so we reformulate the Subspace theorem in the framework of homogeneous dynamics by introducing and studying a slope formalism and the corresponding notion of semistability for diagonal flows. Introduction In 1972, Wolfgang Schmidt formulated his celebrated subspace theorem [32, Lemma 7], a far reaching generalization of results of Thue [37], Siegel [35], and Roth [28] on rational approximations to algebraic numbers. Around the same time, in his work on arithmeticity of lattices in Lie groups, Gregory Margulis [26] used the geometry of numbers to establish the recurrence of unipotent flows on the space of lattices GLd(R)=GLd(Z). More than two decades later, a quantita- tive refinement of this fact, the so-called quantitative non-divergence estimate, was used by Kleinbock and Margulis [18] in their solution to the Sprindzuk conjecture regarding the extremality of non-degenerate manifolds in metric dio- phantine approximation. As it turns out, these two remarkable results – the subspace theorem and the Sprindzuk conjecture – are closely related and can be understood together as statements about diagonal orbits in the space of lat- tices. In this paper we prove a theorem that generalizes both results at the same time. We also provide several applications. This marriage is possible thanks to a better understanding of the geometry lying behind the subspace theorem, in particular the notion of Harder-Narasimhan filtration for one-parameter di- agonal actions, which leads both to a dynamical reformulation of the original subspace theorem and to a further geometric understanding of the family of exceptional subspaces arising in Schmidt’s theorem. -
Mersenne and Fermat Numbers 2, 3, 5, 7, 13, 17, 19, 31
MERSENNE AND FERMAT NUMBERS RAPHAEL M. ROBINSON 1. Mersenne numbers. The Mersenne numbers are of the form 2n — 1. As a result of the computation described below, it can now be stated that the first seventeen primes of this form correspond to the following values of ra: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281. The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127—1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in Archibald [l]; see also Kraitchik [4]. The next five Mersenne primes were found in 1952; they are at present the five largest known primes of any form. They were announced in Lehmer [7] and discussed by Uhler [13]. It is clear that 2" —1 can be factored algebraically if ra is composite; hence 2n —1 cannot be prime unless w is prime. Fermat's theorem yields a factor of 2n —1 only when ra + 1 is prime, and hence does not determine any additional cases in which 2"-1 is known to be com- posite. On the other hand, it follows from Euler's criterion that if ra = 0, 3 (mod 4) and 2ra + l is prime, then 2ra + l is a factor of 2n— 1. -
Primes and Primality Testing
Primes and Primality Testing A Technological/Historical Perspective Jennifer Ellis Department of Mathematics and Computer Science What is a prime number? A number p greater than one is prime if and only if the only divisors of p are 1 and p. Examples: 2, 3, 5, and 7 A few larger examples: 71887 524287 65537 2127 1 Primality Testing: Origins Eratosthenes: Developed “sieve” method 276-194 B.C. Nicknamed Beta – “second place” in many different academic disciplines Also made contributions to www-history.mcs.st- geometry, approximation of andrews.ac.uk/PictDisplay/Eratosthenes.html the Earth’s circumference Sieve of Eratosthenes 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Sieve of Eratosthenes We only need to “sieve” the multiples of numbers less than 10. Why? (10)(10)=100 (p)(q)<=100 Consider pq where p>10. Then for pq <=100, q must be less than 10. By sieving all the multiples of numbers less than 10 (here, multiples of q), we have removed all composite numbers less than 100. -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
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. -
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n.