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Aims the Seminario Matematico Is a Society of Members of Mathematics Aims The Seminario Matematico is a society of members of mathematics-related departments of the University and the Politecnico in Turin. Its scope is to promote study and research in all fields of mathematics and its applications, and to contribute to the diffusion of mathematical culture. The Rendiconti is the official journal of the Seminario Matematico. It publishes papers and invited lectures. Papers should provide original contributions to research, while invited lectures will have a tutorial or overview character. Other information and the contents of recent issues are available at http://seminariomatematico.dm.unito.it/rendiconti/ Instructions to Authors Authors should submit an electronic copy (preferably in the journal’s style, see below) via e-mail: [email protected] English is the preferred language, but papers in Italian or French may also be submitted. The paper should incorporate the name and affiliation of each author, postal and e-mail addresses, a brief abstract, and MSC classification numbers. Authors should inform the executive editor as soon as possible in the event of any change of address. The final decision concerning acceptance of a paper is taken by the editorial board, based upon the evaluation of an external referee. Papers are typically processed within two weeks of arrival, and referees are asked to pass on their reports within two months. Authors should prepare their manuscripts in LATEX, with a separate BIBTEX file whenever possible. To avoid delays in publication, it is essential to provide a LATEX version once a paper has been accepted. The appropriate style file and instructions can be downloaded from the journal’s website. The editorial board reserves the right to refuse publication if the file does not meet these requirements. Reprints are not normally offered. RENDICONTI DEL SEMINARIO MATEMATICO-UNIVERSITÀ E POLITECNICO DI TORINO Università e Politecnico di Torino CONTENTS S. Barbero, U. Cerruti, N. Murru, On Polynomial Solutions of the Diophantine Equation (x + y 1)2 = wxy ........................ 5 − F. Battistoni, Discriminants of number fields and surjectivity of trace homomor- phism on rings of integers .......................... 13 D. Bazzanella and C. Sanna, Least common multiple of polynomial sequences . 21 F. Caldarola, On the maximal finite Iwasawa submodule in Zp-extensions and capitulation of ideals ............................ 27 M. Ceria, T. Mora, M. Sala, Zech tableaux as tools for sparse decoding ..... 43 G. Coppola, Recent results on Ramanujan expansions with applications to cor- relations ................................... 57 M. Elia, Continued Fractions and Factoring ................... 83 E. Tron, The greatest common divisorof linear recurrences ........... 103 Alessandro Gambini, Remis Tonon, Alessandro Zaccagnini, with an addendum by Jacques Benatar and Alon Nishry, Signed harmonic sums of integers with k distinct prime factors ........................ 125 G. Zaghloul, Zeros of generalized Hurwitz zeta functions ............ 143 RENDICONTI DEL SEMINARIO MATEMATICO-UNIVERSITÀ E POLITECNICO DI TORINO EXECUTIVE EDITORS Emilio Musso EDITORIAL COMMITTEE Marino Badiale, Elvise Berchio, Cristiana Bertolin, Erika Luciano, Giovanni Manno, Barbara Trivellato, Maria Vallarino, Ezio Venturino, Domenico Zambella MANAGING COMMITTEE Alessandra De Rossi, Stefano Scialó, Marco Scianna, Lea Terracini, Elena Vigna CONSULTING EDITORS Laurent Desvillettes, Michael Ruzhansky Proprietà letteraria riservata Autorizzazione del Tribunale di Torino N. 2962 del 6.VI.1980 DIRETTORE RESPONSABILE Alberto Collino QUESTO FASCICOLO È PRODOTTO CON IL CONTRIBUTO DI: UNIVERSITÀ DEGLI STUDI DI TORINO POLITECNICO DI TORINO Rendiconti Sem. Mat. Univ. Pol. Torino Vol. 78, 1 (2020), 5 – 12 S. Barbero, U. Cerruti, N. Murru ON POLYNOMIAL SOLUTIONS OF THE DIOPHANTINE EQUATION (X +Y 1)2 = WXY − Abstract. In this paper we consider a particular class of polynomials arising from the solu- tions of the Diophantine equation (x+y 1)2 = wxy. We highlight some interesting aspects, − describing their relationship with many iportant integer sequences and pointing out their con- nection with Dickson and Chebyshev polynomials. We also study their coefficients finding a new identity involving Catalan numbers and proving that they are a Riordan array. 1. A class of polynomials related to integer sequences, Dickson and Chebyshev polynomials In [1], the authors solved the Diophantine equation (1) (x + y 1)2 = wxy, − where w is a given positive integer and x,y are unknown numbers, whose values are to be sought in the set of positive integers. In particular, (x,y) is a solution of the Diophantine equation (1) if and only if (x,y)= +∞ (um+1(w),um(w)), for a given m N, where (un(w)) is the following linear recur- ∈ n=0 rent sequence: u (w)=0, u (w)=1, u (w)=w (2) 0 1 2 !un(w)=(w 1)un 1(w) (w 1)un 2(w)+un 3(w) n 3. − − − − − − ∀ ≥ This polynomial sequence is very interesting. Indeed, for several values of w, the polynomial sequence (un(w)) coincides with some well–known and studied integer 2 sequences. For example, for w = 4, (un(4)) = n , that is the sequence A000290 in OEIS [7]. When w = 5, (un(5)) is the sequence of the alternate Lucas numbers 2 minus 2 (see sequence A004146 in OEIS). If w = 9, (un(9)) = F2n, where (Fn) is the sequence of the Fibonacci numbers. For w = 4,...,20, the sequence (un(w)) ap- pears in OEIS [7]. In Table 1, we summarize sequences un(w) for different values of w. In the following, we prove that polynomials un(w) are related to some well– known and studied polynomials like Chebyshev polynomials of the first and second kind, respectively Tn(x) and Un(x) (see, e.g., [5]), and Dickson polynomials Dn(x) and x En(x)=Un 2 (see, e.g., [3]). Here we define T (x) and U (x) as the n–th element of the linear recurrent sequence " # n n (T (x))+∞ and (U (x))+∞ with characteristic polynomial t2 2xt + 1 and initial con- n n=0 n n=0 − ditions T0(x)=1, T1(x)=x and U0(x)=1, U1(x)=2x, respectively. 5 6 S. Barbero, U. Cerruti, N. Murru +∞ w (un(w))n=0 OEIS reference 2 +∞ 4 0,1,4,9,16,25,... A000290=(n )n=0, 5 0,1,5,16,45,121,... A004146=Alternate Lucas numbers - 2 6 0,1,6,25,96,361,... A092184 7 0,1,7,36,175,841,... A054493 (shifted by one) 8 0,1,8,49,288,1681,... A001108 2 9 0,1,9,64,441,3025,... A049684=F2n (Fn Fibonacci numbers) 10 0,1,10,81,640,5041,..., A095004 (shifted by one) 11 0,1,11,100,891,7921,..., A098296 12 0,1,12,121,1200,11881,... A098297 13 0,1,13,144,1573,17161,... A098298 14 0,1,14,169,2016,24025,... A098299 15 0,1,15,196,2535,32761,... A098300 16 0,1,16,225,3136,43681,... A098301 17 0,1,17,256,3825,57121,... A098302 18 0,1,18,289,4608,73441,... A098303 19 0,1,19,324,5491,93025,... A098304 20 0,1,20,361,6480,116281,... A049683=(L 2)/16 (L Lucas numbers) 6n − n Table 1.1: Sequence un(w) for different values of w We recall that Dickson polynomials are defined as follows: n/2 % & n n i i n 2i D (x)= − ( 1) x − n ∑ n i i − i=0 − $ % and n/2 % & n i i n 2i E (x)= − ( 1) x − . n ∑ i − i=0 $ % We also recall that for Dickson polynomials the following identities hold n+1 (n+1) 1 n n 1 x x− (3) D x + x− = x + x− , E x + x− = − n n x x 1 − − " # " # THEOREM 1. We have w 2 D (w 2) 2 Tn( − ) 1 (4) u (w)= n − − = 2 2 − , n 0 n w 4 w 4 ∀ ≥ − − and in particular for all n 1 ≥ 2 2 w 2 (5) u2n(w)=wEn 1(w 2)=wUn 1 − − − − 2 $ % 2 u2n 1(w)=(En 1(w 2)+En 2(w 2)) = − − − − − (6) w 2 w 2 2 = Un 1 − +Un 2 − − 2 − 2 $ $ % $ %% On Polynomial Solutions of the Diophantine Equation (x + y 1)2 = wxy 7 − Proof. The recurrence relation described in (2) clearly shows that the characteristic polynomial of (un(w)) is x3 (w 1)x2 +(w 1)x 1 =(x 1)(x2 (w 2)x + 1) − − − − − − − w 2 √w2 4w whose zeros are x1 = 1 and x2,3 = − ± 2 − . If we set x2 = ζ we easily observe that x = ζ 1 so that ζ + ζ 1 = w 2 and ζ ζ 1 = √w2 4w. Moreover, using the 3 − − − − − − initial conditions in (2), with standard tecniques we find the following closed form for every element of (un(w)) ζn + ζ n 2 ζn + ζ n 2 (7) u (w)= − − = − − n w 4 ζ + ζ 1 2 − − − . Thanks to the first identity in (3) it is straightforward to observe that D (ζ + ζ 1) 2 D (w 2) 2 (8) u (w)= n − − = n − − . n w 4 w 4 − − 2 w 2 Since x (w 2)x + 1 is the characteristic polynomial of the sequence (T − ), − − n 2 with roots x = ζ and x = ζ 1, and the initial conditions are T w 2 = 1, T w 2 = 2 3 − 0 −2 1 " −2 # w 2 we obtain −2 " # " # w 2 ζn + ζ n D (ζ + ζ 1) D (w 2) (9) T − = − = n − = n − n 2 2 2 2 $ % Thus substituting (9) in (8) we prove equality (4). Now considering the equality (7) and the second identity in (3) we have 2n 2n n n 2 1 2 ζ + ζ− 2 (ζ ζ− ) (ζ ζ− ) 2 u2n(w)= − = − − = w(En 1(w 2)) , ζ + ζ 1 2 (ζ ζ 1)2 ζ + ζ 1 2 − − − − − − − − which proves (5), and 2n 1 2n+1 2n 1 2n+1 1 ζ − + ζ− 2 (ζ − + ζ− 2)(ζ + ζ− + 2) (10) u2n 1(w)= − = − − ζ + ζ 1 2 (ζ ζ 1)2 − − − − where we use the identity 1 2 1 1 (ζ ζ− ) = w(w 4)=(ζ + ζ− + 2)(ζ + ζ− 2).
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