In memoriam Péter Kiss
|
|
- Elinor Johnston
- 5 years ago
- Views:
Transcription
1 Kálmán Liptai Eszterházy Károly Collage, Leányka út 4, 3300 Eger, Hungary (Submitted March 2002-Final Revision October 2003) In memoriam Péter Kiss ABSTRACT A positive integer n is called a balancing number if (n 1) = (n + 1) + (n + 2) + + (n + r) for some natural number r. We prove that there is no Fibonacci balancing number except INTRODUCTION The sequence {R n } n=0 = R(A, B, R 0, R 1 ) is called a second order linear recurrence if the recurrence relation R n = AR n 1 + BR n 2 (n > 1) holds for its terms, where A, B 0, R 0 and R 1 are fixed rational integers and R 0 + R 1 > 0. The polynomial x 2 Ax B is called the companion polynomial of the second order linear recurrence sequence R = R(A, B, R 0, R 1 ). The zeros of the companion polynomial will be denoted by α and β. Using this notation, as it is well known, we get where a = R 1 R 0 β and b = R 1 R 0 α (see [6]). A positive integer n is called a balancing number [3] if R n = aαn bβ n, (1) α β (n 1) = (n + 1) + (n + 2) + + (n + r) for some r Z +. Here r is called the balancer corresponding to the balancing number n. For example 6 and 3 are balancing numbers with balancers 2 and 14. In a joint paper A. Behera and G. K. Panda [3] proved that the balancing numbers fulfil the following recurrence relation B n+1 = 6B n B n 1 (n > 1) (2) where B 0 = 1 and B 1 = 6. We call a balancing number a Fibonacci balancing number if it is a Fibonacci number, too. In the next section we prove that there are no Fibonacci balancing numbers. 2. The equation x 2 Dy 2 = N with given integers D and N and variables x and y, is called Pell s equation. First we prove that the balancing numbers are solutions of a Pell s equation. 330
2 Theorem 1: The terms of the second order linear recurrence B(6, 1, 1, 6) are the solutions of the equation z 2 y 2 = 1 (3) for some integer z. Proof: Let B(6, 1, 1, 6) be a second order linear recurrence and denote by α and β the zeros of their companion polynomials and D the discriminant of the companion polynomial. Using (1) and the definition of α and β we get therefore, with y = B n, we have B n = ( )α n (3 2 2)β n 4 2 and αβ = 1, 1 + y 2 = 1 + Bn 2 = 1 + ( ( ) 2 α 2n 32 2( )(3 2 2)α n β n + (3 2 2) 2 β 2n) ( (3 + 2 ) 2 ( 2)α n = (3 2 2)β n 2 = ( ( )α n 2 ) 2 + (3 2 2)β n = z 2. 2 ) 2 Using that α = , β = and the binomial formula it can be proved that z is a rational integer. To prove our main result we need the following theorem of P. E. Ferguson [4]. Theorem 2: The only solutions of the equation x 2 y 2 = ±4 (4) are x = ±L n, y = ±F n (n = 0, 1, 2,... ), where L n and F n are the n th terms of the Lucas and Fibonacci sequences, respectively. Using the method of A. Baker and H. Davenport we prove that there are finitely many common solutions of the Pell s equations (3) and (4). We remark that this result follows from a theorem of P. Kiss [], too. In the process we show that there are no Fibonacci balancing numbers. In the proof we use the following theorem of A. Baker and H. Wüstholz [2]. Theorem 3: Let α 1,..., α n be algebraic numbers not 0 or 1, and let Λ = b 1 log α b n log α n, where b 1,..., b n are rational integers not all zeros. 331
3 We suppose that B = max( b 1,..., b k, e) and A i = max{(h(α i ), e} (i = 1, 2,... n). Assume that the field K generated by α 1, α 2,..., α n over the rationals has degree at most d. If Λ 0 then log Λ > (16nd) 2(n+2) log A 1 log A 2... log A n log B. (H(α) is equal to the maximum of absolute values of the coefficients of the minimal defining polynomial of α.) The following theorem is the main result of this paper. Theorem 4: There is no Fibonacci balancing number except 1. Proof: First we show that there are finitely many common solutions of the equations (, (6) and ( ), (6 ) and x = y 2 ( x 2 4 = y 2 ( ) x = z 2 (6) x = z 2 (6 ) The equations ( and ( ) can be written as (y + x (y x = 4 (7) (y + x (y x = 4. () y + x = (y 0 + x 0 (9 + 4 m where m 0, it is easily verified (by combining this equation with its conjugate) that y 0 is always positive but x 0 is negative if m is large. Hence we can choose m so that x 0 > 0; but if x 1 is defined by y 0 + x 0 = (y1 + x 1 (9 + 4 then x 1 0. Since y 0 + x 0 = (9y1 + 20x 1 ) + (9x 1 + 4y 1 ) we have y 0 = 9y x 1 and x 0 = 9x 1 + 4y 1. From the previous equations we have x 1 = 9x 0 4y 0 and x 0 4y 0 9. Using equation ( we have y = x y2 0. Hence y 0 = 3, 7, 1 and x 0 = 1, 3,, respectively. Thus the general solution of equation ( is given by y + x = (3 + (9 + 4 m (9) y + x = (7 + 3 (9 + 4 m (10) where m = 0, 1, 2... y + x = (1 + (9 + 4 m (11) 332
4 Using the same method as before with ( ), we find that y 1 = 9y 0 20x 0 0 (in this case x 0 is always positive), whence y0 2 = x x2 0, so that x 0 = 1, 2, and y 0 = 1, 4, 11, respectively. Thus the general solution of equation ( ) is given by y + x = (1 + (9 + 4 m (12) y + x = (4 + 2 (9 + 4 m (13) y + x = (11 + (9 + 4 m (14) where m = 0, 1, 2,... The general solution of equations (6) and (6 ) is given by z + x = (3 + ) n (1 where n = 0, 1, 2,... We are looking for the common solutions of the equation (9), (10), (11), (12), (13), (14) with the equations (1. Using (9), (1 and their conjugates we have 2x = (3 + ) n (3 ) n = 3 + (9 + 4 m 3 (9 4 m and so 1 (3 + ) n (3 + ) n = + 3 ( m + (9 + 4 m. (16) Putting Q = 1 (3 + ) n, P = + 3 (9 + 4 m, in equation (16) we obtain Q 1 Q 1 = P 4 P 1. (17) Since Q P = 1 Q 1 4 P 1 < 4 (Q 1 P 1 ) = 4 P Q QP and plainly P > 1 and Q > 1, we have Q < P. Also P Q = 4 P 1 1 Q 1 < 4 P 1 and P >
5 0 < log P ( Q = log 1 P Q ) < 4 ( ) 2 4 P P 2 + P 2 = 4 P P 4 < 0.1P 2 < 0.1 ( Using the previous inequality and the definitions of P and Q, we get 0 < m log(9 + 4 n log(3 + ) + log (3 + < We apply Theorem 3 with n = 3 and α 1 = α 2 = 3 + α 3 = ( ( (1) We use that 0.1(( ) m < e.77m. The equations satisfied by α 1, α 2, α 3 are α 2 1 1α = 0 α 2 2 6α = 0 2α α = 0. Hence A 1 = 1, A 2 = 6, A 3 = 1120 and d = 4. Using Theorem 3 and the previous inequality we have.77 (16 3 4)10 log 1 log 6 log 1120 log 0 24 log m. Thus we have Using the same method we investigate the equations (10) and (1. We have 2x = (9 + 4 m 7 3 (9 4 m = (3 + ) n (3 ) n that is (9 + 4 m 7 3 (9 + 4 m = (3 + ) n (3 + ) n. (19) P 1 = (9 + 4 m, Q = 1 (3 + ) n (20) 334
6 using (19) and (20) we have Using the previous method we have Q 1 Q 1 = P 1 4 P < log P 1 Q < 0.1P 2 1 < ( Substituting from (20), we obtain 0 < m log(9 + 4 n log(3 + ) + log (7 + 3 < We apply Theorem 3 with n = 3 and α 1 = α 2 = 3 + α 3 = ( ( (21) The equation 2α α = 0 is satisfied by α 3, that is A 3 = 720. Using Theorem 3 as above we have.77 (16 3 4)10 log 1 log 6 log 720 log 0 24 log m From the equations (11) and (1 we have 2x = 1 + (9 + 4 m 1 (9 4 m = (3 + ) n (3 ) n that is 1 + (9 + 4 m 1 (9 + 4 m = (3 + ) n (3 + ) n. (22) P 2 = 1 + (9 + 4 m, Q = (3 + ) n. (23) 33
7 Using (22) and (23) we have Q 1 Q 1 = P 2 4 P 1 2. As before we obtain Substituting from (23) we have 0 < log P ( 2 Q = log 1 P ) 2 Q < P ( < m log(9 + 4 n log(3 + ) + log (1 + < We apply Theorem 3 as above with n = 3 and α 1 = α 2 = 3 + α 3 = ( ( (24) The equation 2α 4 120α = 0 is satisfied by α 3 and so A 3 = (16 3 4)10 log 1 log 6 log 120 log 0 24 log m From the equations (12) and (1 we have 2x = 1 + (9 + 4 m 1 (9 4 m = (3 + ) n (3 ) n that is 1 + (9 + 4 m 1 (9 + 4 m = (3 + ) n (3 + ) n. (2 P 3 = 1 + (9 + 4 m, Q = (3 + ) n. (26) 336
8 Using (2 and (26) we have From the previous equation we have Q 1 Q 1 = P P 1 3. that is Q > P 3 and Q P 3 = 4 P Q 1 > 0 Q P 3 < 4 P P 1 3 = P < log Q ( = log 1 + Q P ) 3 < 37 P 3 P 3 40 P 3 2 and so ( ) P 3 2 < 0.926P3 2 1 < ( < m log(9 + 4 n log(3 + ) + log Using the previous method and that the equation is satisfied by α 3 = 1024α 4 40α = 0 (1+ and so A 3 = (1 + < ( (27).77 (16 3 4)10 log 1 log 6 log 1024 log 0 24 log m From the equations (13) and (1 we have (9 + 4 m 4 2 (9 4 m = (3 + ) n (3 ) n. 337
9 we get similar inequalities as before. We have P 4 = (9 + 4 m, Q = (3 + ) n and 0 < log Q < 0.926P < 0.06 P 4 ( < m log(9 + 4 n log(3 + ) + log (4 + 2 < ( (2) The equation 62464α 4 20α 2 + = 0 is satisfied by α 3 = (4+2 and so A 3 = We use the Theorem of A. Baker and G. Wüstholz again and we have.77 (16 3 4)10 log 1 log 6 log log 0 24 log m Finally let s consider the equations (14) and (1. We have 11 + (9 + 4 m 11 (9 4 m = (3 + ) n (3 ) n. and we use the previous steps we have P = 11 + (9 + 4 m, Q = (3 + ) n and 0 < log Q < 0.926P 2 1 < P ( < m log(9 + 4 n log(3 + ) + log (11 + < ( (29) The equation 1024α α = 0 33
10 is satisfied by α 3 = (11+ and so A 3 = We apply again Theorem 3 and we have.77 (16 3 4)10 log 1 log 6 log 1960 log 0 24 log m We get that there are finitely many common solutions of simultaneous equations, that is, there are finitely many Fibonacci balancing numbers. Since the bounds for m are too high (0 26 ) we can t investigate all of them. In order to get a lower bound we use the following lemma of A. Baker and H. Davenport [1]. Lemma: Suppose that K > 6. For any positive integer M, let p and q be integers satisfying 1 q KM, θq p < 2(KM) 1. Then, if qβ 3K 1, there is no solution of the equation in the range mθ n + β < C m log K 2 M log C < m < M. (It is supposed that θ, β are real numbers and C > 1. z denotes the distance of a real number z from the nearest integer.) We divide the inequality (1) by log(3 + ) and we show the steps of reduction. In the other cases, (21), (24), (27), (2) and (29), the method is similar. Using the lemma we have and C = ( = , θ = log(9 + 4 log(3 + ) β = log (3 + (log(3 + )) 1. In our case we take M = 10 26, K = 100. Let θ 0 be the value of θ correct to 6 decimal places, so that θ θ 0 < Let p q be the last convergent to the continued fraction for θ 0 which satisfies q < 10 2 ; then qθ 0 p < We therefore have qθ p q θ θ 0 + qθ 0 p < In this case the first and the second inequalities of the Lemma are satisfied. The values of θ, β and q computed by Maple are given in Appendix. We have that qβ = It follows from the lemma, since qβ 0.03, there is no solution of (1) in the range log log 321, 997 <
11 That is m less than 12, so we can calculate by hand that there is no Fibonacci balancing number in this case. We get the same result in the other cases, too. ACKNOWLEDGMENTS I would like to thank Professor L. Hajdú and Á. Reszegi for their valuable and useful advices. Research supported by the Hungarian OTKA Foundation No. T 0329 and the György Békésy Foudation. APPENDIX θ = q = β = qβ = β 1 = qβ 1 = β 2 = qβ 2 = β 3 = qβ 3 = β 4 = qβ 4 = β = qβ = REFERENCES [1] A. Baker and H. Davenport. The Equations 3x 2 2 = y 2 and x 2 7 = z 2. Quart. J. Math. Oxford 20.2 (1969): [2] A. Baker and G. Wüstholz. Logarithmic Forms and Group Varieties. J. Reine Angew. Math. 442 (1993): [3] A. Behera and G. K. Panda. On the Square Roots of Triangular Numbers. The Fibonacci Quarterly 37.2 (1999): [4] D. E. Ferguson. Letter to the editor. The Fibonacci Quarterly (1970): 9. [] P. Kiss. On Common Terms of Linear Recurrences. Acta Math. Acad. Sci. Hungar 40 (192): [6] T. N. Shorey and R. Tijdeman. Exponential Diophantine Equations. Cambridge University Press (196). AMS Classification Numbers: 11B39, 11D4 340
ON GENERALIZED BALANCING SEQUENCES
ON GENERALIZED BALANCING SEQUENCES ATTILA BÉRCZES, KÁLMÁN LIPTAI, AND ISTVÁN PINK Abstract. Let R i = R(A, B, R 0, R 1 ) be a second order linear recurrence sequence. In the present paper we prove that
More informationOn the Shifted Product of Binary Recurrences
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 13 (2010), rticle 10.6.1 On the Shifted Product of Binary Recurrences Omar Khadir epartment of Mathematics University of Hassan II Mohammedia, Morocco
More informationSome congruences concerning second order linear recurrences
Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae,. (1997) pp. 9 33 Some congruences concerning second order linear recurrences JAMES P. JONES PÉTER KISS Abstract. Let U n V n (n=0,1,,...) be
More informationOn repdigits as product of consecutive Lucas numbers
Notes on Number Theory and Discrete Mathematics Print ISSN 1310 5132, Online ISSN 2367 8275 Vol. 24, 2018, No. 3, 5 102 DOI: 10.7546/nntdm.2018.24.3.5-102 On repdigits as product of consecutive Lucas numbers
More informationFurther generalizations of the Fibonacci-coefficient polynomials
Annales Mathematicae et Informaticae 35 (2008) pp 123 128 http://wwwektfhu/ami Further generalizations of the Fibonacci-coefficient polynomials Ferenc Mátyás Institute of Mathematics and Informatics Eszterházy
More informationON THE EXTENSIBILITY OF DIOPHANTINE TRIPLES {k 1, k + 1, 4k} FOR GAUSSIAN INTEGERS. Zrinka Franušić University of Zagreb, Croatia
GLASNIK MATEMATIČKI Vol. 43(63)(2008), 265 291 ON THE EXTENSIBILITY OF DIOPHANTINE TRIPLES {k 1, k + 1, 4k} FOR GAUSSIAN INTEGERS Zrinka Franušić University of Zagreb, Croatia Abstract. In this paper,
More informationA note on the products of the terms of linear recurrences
Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 24 (1997) pp 47 53 A note on the products of the terms of linear recurrences LÁSZLÓ SZALAY Abstract For an integer ν>1 let G (i) (,,ν) be linear
More informationA parametric family of quartic Thue equations
A parametric family of quartic Thue equations Andrej Dujella and Borka Jadrijević Abstract In this paper we prove that the Diophantine equation x 4 4cx 3 y + (6c + 2)x 2 y 2 + 4cxy 3 + y 4 = 1, where c
More informationActa Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 21 26
Acta Acad. Paed. Agriensis, Sectio Mathematicae 8 (00) 6 APPROXIMATION BY QUOTIENTS OF TERMS OF SECOND ORDER LINEAR RECURSIVE SEQUENCES OF INTEGERS Sándor H.-Molnár (Budapest, Hungary) Abstract. In the
More informationON POWER VALUES OF POLYNOMIALS. A. Bérczes, B. Brindza and L. Hajdu
ON POWER VALUES OF POLYNOMIALS ON POWER VALUES OF POLYNOMIALS A. Bérczes, B. Brindza and L. Hajdu Abstract. In this paper we give a new, generalized version of a result of Brindza, Evertse and Győry, concerning
More informationOn Some Combinations of Non-Consecutive Terms of a Recurrence Sequence
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 21 (2018), Article 18.3.5 On Some Combinations of Non-Consecutive Terms of a Recurrence Sequence Eva Trojovská Department of Mathematics Faculty of Science
More informationOn arithmetic functions of balancing and Lucas-balancing numbers
MATHEMATICAL COMMUNICATIONS 77 Math. Commun. 24(2019), 77 1 On arithmetic functions of balancing and Lucas-balancing numbers Utkal Keshari Dutta and Prasanta Kumar Ray Department of Mathematics, Sambalpur
More informationFlorian Luca Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P , Morelia, Michoacán, México
Florian Luca Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 8180, Morelia, Michoacán, México e-mail: fluca@matmor.unam.mx Laszlo Szalay Department of Mathematics and Statistics,
More informationFibonacci numbers of the form p a ± p b
Fibonacci numbers of the form p a ± p b Florian Luca 1 and Pantelimon Stănică 2 1 IMATE, UNAM, Ap. Postal 61-3 (Xangari), CP. 8 089 Morelia, Michoacán, Mexico; e-mail: fluca@matmor.unam.mx 2 Auburn University
More informationParallel algorithm for determining the small solutions of Thue equations
Annales Mathematicae et Informaticae 40 (2012) pp. 125 134 http://ami.ektf.hu Parallel algorithm for determining the small solutions of Thue equations Gergő Szekrényesi University of Miskolc, Department
More informationarxiv: v4 [math.nt] 3 Jul 2017
PRIME POWERS IN SUMS OF TERMS OF BINARY RECURRENCE SEQUENCES arxiv:161002774v4 [mathnt] 3 Jul 2017 ESHITA MAZUMDAR AND S S ROUT Abstract Let {u n } n 0 be a non-degeneratebinaryrecurrencesequence with
More informationGeneralized balancing numbers
Indag. Mathem., N.S., 20 (1, 87 100 March, 2009 Generalized balancing numbers by Kálmán Liptai a, Florian Luca b,ákospintér c and László Szalay d a Institute of Mathematics and Informatics, Eszterházy
More informationRECIPROCAL SUMS OF SEQUENCES INVOLVING BALANCING AND LUCAS-BALANCING NUMBERS
RECIPROCAL SUMS OF SEQUENCES INVOLVING BALANCING AND LUCAS-BALANCING NUMBERS GOPAL KRISHNA PANDA, TAKAO KOMATSU and RAVI KUMAR DAVALA Communicated by Alexandru Zaharescu Many authors studied bounds for
More informationNEW IDENTITIES FOR THE COMMON FACTORS OF BALANCING AND LUCAS-BALANCING NUMBERS
International Journal of Pure and Applied Mathematics Volume 85 No. 3 013, 487-494 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.173/ijpam.v85i3.5
More informationPower values of polynomials and binomial Thue Mahler equations
Publ. Math. Debrecen 65/3-4 (2004), 341 362 Power values of polynomials and binomial Thue Mahler equations By K. GYŐRY (Debrecen), I. PINK (Debrecen) and A. PINTÉR (Debrecen) To the memory of Professors
More informationTHE NUMBER OF DIOPHANTINE QUINTUPLES. Yasutsugu Fujita College of Industrial Technology, Nihon University, Japan
GLASNIK MATEMATIČKI Vol. 45(65)(010), 15 9 THE NUMBER OF DIOPHANTINE QUINTUPLES Yasutsugu Fujita College of Industrial Technology, Nihon University, Japan Abstract. A set a 1,..., a m} of m distinct positive
More informationA family of quartic Thue inequalities
A family of quartic Thue inequalities Andrej Dujella and Borka Jadrijević Abstract In this paper we prove that the only primitive solutions of the Thue inequality x 4 4cx 3 y + (6c + 2)x 2 y 2 + 4cxy 3
More informationON THE REPRESENTATION OF FIBONACCI AND LUCAS NUMBERS IN AN INTEGER BASES
ON THE REPRESENTATION OF FIBONACCI AND LUCAS NUMBERS IN AN INTEGER BASES YANN BUGEAUD, MIHAI CIPU, AND MAURICE MIGNOTTE À Paulo Ribenboim, pour son quatre-vingtième anniversaire Abstract. Résumé. Nous
More informationCullen Numbers in Binary Recurrent Sequences
Cullen Numbers in Binary Recurrent Sequences Florian Luca 1 and Pantelimon Stănică 2 1 IMATE-UNAM, Ap. Postal 61-3 (Xangari), CP 58 089 Morelia, Michoacán, Mexico; e-mail: fluca@matmor.unam.mx 2 Auburn
More informationMathematica Slovaca. Jaroslav Hančl; Péter Kiss On reciprocal sums of terms of linear recurrences. Terms of use:
Mathematica Slovaca Jaroslav Hančl; Péter Kiss On reciprocal sums of terms of linear recurrences Mathematica Slovaca, Vol. 43 (1993), No. 1, 31--37 Persistent URL: http://dml.cz/dmlcz/136572 Terms of use:
More informationDiophantine quadruples and Fibonacci numbers
Diophantine quadruples and Fibonacci numbers Andrej Dujella Department of Mathematics, University of Zagreb, Croatia Abstract A Diophantine m-tuple is a set of m positive integers with the property that
More informationCombinatorial Diophantine Equations
Combinatorial Diophantine Equations Szabolcs Tengely Arctic Number Theory School, Helsinki May 21, 2011 Integer sequences Problem: find intersection of integer sequences Some well-known sequences: perfect
More informationSIMULTANEOUS PELL EQUATIONS. Let R and S be positive integers with R < S. We shall call the simultaneous Diophantine equations
MATHEMATICS OF COMPUTATION Volume 65 Number 13 January 1996 Pages 355 359 SIMULTANEOUS PELL EQUATIONS W. S. ANGLIN Abstract. Let R and S be positive integers with R
More informationCertain Diophantine equations involving balancing and Lucas-balancing numbers
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 0, Number, December 016 Available online at http://acutm.math.ut.ee Certain Diophantine equations involving balancing and Lucas-balancing
More informationPowers of Two as Sums of Two Lucas Numbers
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 17 (014), Article 14.8.3 Powers of Two as Sums of Two Lucas Numbers Jhon J. Bravo Mathematics Department University of Cauca Street 5 No. 4-70 Popayán,
More informationTwo Diophantine Approaches to the Irreducibility of Certain Trinomials
Two Diophantine Approaches to the Irreducibility of Certain Trinomials M. Filaseta 1, F. Luca 2, P. Stănică 3, R.G. Underwood 3 1 Department of Mathematics, University of South Carolina Columbia, SC 29208;
More informationBalancing And Lucas-balancing Numbers With Real Indices
Balancing And Lucas-balancing Numbers With Real Indices A thesis submitted by SEPHALI TANTY Roll No. 413MA2076 for the partial fulfilment for the award of the degree Master Of Science Under the supervision
More informationON `-TH ORDER GAP BALANCING NUMBERS
#A56 INTEGERS 18 (018) ON `-TH ORDER GAP BALANCING NUMBERS S. S. Rout Institute of Mathematics and Applications, Bhubaneswar, Odisha, India lbs.sudhansu@gmail.com; sudhansu@iomaorissa.ac.in R. Thangadurai
More information#A2 INTEGERS 12A (2012): John Selfridge Memorial Issue ON A CONJECTURE REGARDING BALANCING WITH POWERS OF FIBONACCI NUMBERS
#A2 INTEGERS 2A (202): John Selfridge Memorial Issue ON A CONJECTURE REGARDING BALANCING WITH POWERS OF FIBONACCI NUMBERS Saúl Díaz Alvarado Facultad de Ciencias, Universidad Autónoma del Estado de México,
More informationABSTRACT 1. INTRODUCTION
THE FIBONACCI NUMBER OF GENERALIZED PETERSEN GRAPHS Stephan G. Wagner Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria e-mail: wagner@finanz.math.tu-graz.ac.at
More informationOn integer solutions to x 2 dy 2 = 1, z 2 2dy 2 = 1
ACTA ARITHMETICA LXXXII.1 (1997) On integer solutions to x 2 dy 2 = 1, z 2 2dy 2 = 1 by P. G. Walsh (Ottawa, Ont.) 1. Introduction. Let d denote a positive integer. In [7] Ono proves that if the number
More informationPILLAI S CONJECTURE REVISITED
PILLAI S COJECTURE REVISITED MICHAEL A. BEETT Abstract. We prove a generalization of an old conjecture of Pillai now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3 x
More informationActa Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) THE LIE AUGMENTATION TERMINALS OF GROUPS. Bertalan Király (Eger, Hungary)
Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 93 97 THE LIE AUGMENTATION TERMINALS OF GROUPS Bertalan Király (Eger, Hungary) Abstract. In this paper we give necessary and sufficient conditions
More informationFIBONACCI DIOPHANTINE TRIPLES. Florian Luca and László Szalay Universidad Nacional Autonoma de México, Mexico and University of West Hungary, Hungary
GLASNIK MATEMATIČKI Vol. 436300, 53 64 FIBONACCI DIOPHANTINE TRIPLES Florian Luca and László Szalay Universidad Nacional Autonoma de México, Mexico and University of West Hungary, Hungary Abstract. In
More informationPELLANS SEQUENCE AND ITS DIOPHANTINE TRIPLES. Nurettin Irmak and Murat Alp
PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 100114 016, 59 69 DOI: 10.98/PIM161459I PELLANS SEQUENCE AND ITS DIOPHANTINE TRIPLES Nurettin Irmak and Murat Alp Abstract. We introduce a novel
More informationTo Professor W. M. Schmidt on his 60th birthday
ACTA ARITHMETICA LXVII.3 (1994) On the irreducibility of neighbouring polynomials by K. Győry (Debrecen) To Professor W. M. Schmidt on his 60th birthday 1. Introduction. Denote by P the length of a polynomial
More informationOn prime factors of subset sums
On prime factors of subset sums by P. Erdös, A. Sárközy and C.L. Stewart * 1 Introduction For any set X let X denote its cardinality and for any integer n larger than one let ω(n) denote the number of
More informationDIGITAL EXPANSION OF EXPONENTIAL SEQUENCES
DIGITAL EXPASIO OF EXPOETIAL SEQUECES MICHAEL FUCHS Abstract. We consider the q-ary digital expansion of the first terms of an exponential sequence a n. Using a result due to Kiss und Tichy [8], we prove
More informationDiophantine m-tuples and elliptic curves
Diophantine m-tuples and elliptic curves Andrej Dujella (Zagreb) 1 Introduction Diophantus found four positive rational numbers 1 16, 33 16, 17 4, 105 16 with the property that the product of any two of
More informationTHE EQUATIONS 3z 2-2 = y 2 AND &c 2-7 = z 2
THE EQUATIONS 3z 2-2 = y 2 AND &c 2-7 = z 2 5y A. BAKER and H. DAVENPORT [Received 6 November 1968] 1. THE four numbers 1, 3, 8, 120 have the property that the product of any two, increased by 1, is a
More informationPOWER VALUES OF SUMS OF PRODUCTS OF CONSECUTIVE INTEGERS. 1. Introduction. (x + j).
POWER VALUES OF SUMS OF PRODUCTS OF CONSECUTIVE INTEGERS L. HAJDU, S. LAISHRAM, SZ. TENGELY Abstract. We investigate power values of sums of products of consecutive integers. We give general finiteness
More informationPower values of sums of products of consecutive integers
isid/ms/2015/15 October 12, 2015 http://www.isid.ac.in/ statmath/index.php?module=preprint Power values of sums of products of consecutive integers L. Hajdu, S. Laishram and Sz. Tengely Indian Statistical
More informationAlmost fifth powers in arithmetic progression
Almost fifth powers in arithmetic progression L. Hajdu and T. Kovács University of Debrecen, Institute of Mathematics and the Number Theory Research Group of the Hungarian Academy of Sciences Debrecen,
More informationOn the resolution of simultaneous Pell equations
Annales Mathematicae et Informaticae 34 (2007) pp. 77 87 http://www.ektf.hu/tanszek/matematika/ami On the resolution of simultaneous Pell equations László Szalay Institute of Mathematics and Statistics,
More information1 Diophantine quintuple conjecture
Conjectures and results on the size and number of Diophantine tuples Andrej Dujella Department of Mathematics, University of Zagreb Bijenička cesta 30, 10000 Zagreb, CROATIA E-mail: duje@math.hr URL :
More informationEFFECTIVE SOLUTION OF THE D( 1)-QUADRUPLE CONJECTURE
EFFECTIVE SOLUTION OF THE D( 1)-QUADRUPLE CONJECTURE ANDREJ DUJELLA, ALAN FILIPIN, AND CLEMENS FUCHS Abstract. The D( 1)-quadruple conjecture states that there does not exist a set of four positive integers
More informationLINEAR RECURRENCES AND CHEBYSHEV POLYNOMIALS
LINEAR RECURRENCES AND CHEBYSHEV POLYNOMIALS Sergey Kitaev Matematik Chalmers tekniska högskola och Göteborgs universitet S-412 96 Göteborg Sweden e-mail: kitaev@math.chalmers.se Toufik Mansour Matematik
More informationSolving binomial Thue equations
arxiv:1810.01819v1 [math.nt] 27 Sep 2018 Solving binomial Thue equations István Gaál and László Remete University of Debrecen, Mathematical Institute H 4010 Debrecen Pf.12., Hungary e mail: igaal@science.unideb.hu,
More informationA remark about the positivity problem of fourth order linear recurrence sequences
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 18, Number 1, June 2014 Available online at http://acutm.math.ut.ee A remark about the positivity problem of fourth order linear recurrence
More informationON A THEOREM OF TARTAKOWSKY
ON A THEOREM OF TARTAKOWSKY MICHAEL A. BENNETT Dedicated to the memory of Béla Brindza Abstract. Binomial Thue equations of the shape Aa n Bb n = 1 possess, for A and B positive integers and n 3, at most
More informationFifteen problems in number theory
Acta Univ. Sapientiae, Mathematica, 2, 1 (2010) 72 83 Fifteen problems in number theory Attila Pethő University of Debrecen Department of Computer Science and Number Theory Research Group Hungarian Academy
More informationBalancing sequences of matrices with application to algebra of balancing numbers
Notes on Number Theory and Discrete Mathematics ISSN 1310 5132 Vol 20 2014 No 1 49 58 Balancing sequences of matrices with application to algebra of balancing numbers Prasanta Kumar Ray International Institute
More informationLINEAR FORMS IN LOGARITHMS
LINEAR FORMS IN LOGARITHMS JAN-HENDRIK EVERTSE April 2011 Literature: T.N. Shorey, R. Tijdeman, Exponential Diophantine equations, Cambridge University Press, 1986; reprinted 2008. 1. Linear forms in logarithms
More informationINCOMPLETE BALANCING AND LUCAS-BALANCING NUMBERS
INCOMPLETE BALANCING AND LUCAS-BALANCING NUMBERS BIJAN KUMAR PATEL, NURETTIN IRMAK and PRASANTA KUMAR RAY Communicated by Alexandru Zaharescu The aim of this article is to establish some combinatorial
More informationBILGE PEKER, ANDREJ DUJELLA, AND SELIN (INAG) CENBERCI
THE NON-EXTENSIBILITY OF D( 2k + 1)-TRIPLES {1, k 2, k 2 + 2k 1} BILGE PEKER, ANDREJ DUJELLA, AND SELIN (INAG) CENBERCI Abstract. In this paper we prove that for an integer k such that k 2, the D( 2k +
More informationOn Generalized Fibonacci Polynomials and Bernoulli Numbers 1
1 3 47 6 3 11 Journal of Integer Sequences, Vol 8 005), Article 0553 On Generalized Fibonacci Polynomials and Bernoulli Numbers 1 Tianping Zhang Department of Mathematics Northwest University Xi an, Shaanxi
More informationFibonacci Diophantine Triples
Fibonacci Diophantine Triples Florian Luca Instituto de Matemáticas Universidad Nacional Autonoma de México C.P. 58180, Morelia, Michoacán, México fluca@matmor.unam.mx László Szalay Institute of Mathematics
More informationThe Diophantine equation x n = Dy 2 + 1
ACTA ARITHMETICA 106.1 (2003) The Diophantine equation x n Dy 2 + 1 by J. H. E. Cohn (London) 1. Introduction. In [4] the complete set of positive integer solutions to the equation of the title is described
More informationTHE PROBLEM OF DIOPHANTUS FOR INTEGERS OF. Zrinka Franušić and Ivan Soldo
THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF Q( ) Zrinka Franušić and Ivan Soldo Abstract. We solve the problem of Diophantus for integers of the quadratic field Q( ) by finding a D()-quadruple in Z[( + )/]
More informationON CERTAIN COMBINATORIAL DIOPHANTINE EQUATIONS AND THEIR CONNECTION TO PYTHAGOREAN NUMBERS
ON CERTAIN COMBINATORIAL DIOPHANTINE EQUATIONS AND THEIR CONNECTION TO PYTHAGOREAN NUMBERS ROBERT S. COULTER, MARIE HENDERSON, AND FELIX LAZEBNIK 1. Introduction The binomial knapsack problem is easily
More informationActa Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 19 24
Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 19 24 PRIMITIVE DIVISORS OF LUCAS SEQUENCES AND PRIME FACTORS OF x 2 + 1 AND x 4 + 1 Florian Luca (Michoacán, México) Abstract. In this
More informationBOUNDS FOR DIOPHANTINE QUINTUPLES. Mihai Cipu and Yasutsugu Fujita IMAR, Romania and Nihon University, Japan
GLASNIK MATEMATIČKI Vol. 5070)2015), 25 34 BOUNDS FOR DIOPHANTINE QUINTUPLES Mihai Cipu and Yasutsugu Fujita IMAR, Romania and Nihon University, Japan Abstract. A set of m positive integers {a 1,...,a
More informationRepresenting integers as linear combinations of powers
ubl. Math. Debrecen Manuscript (August 15, 2011) Representing integers as linear combinations of powers By Lajos Hajdu and Robert Tijdeman Dedicated to rofessors K. Győry and A. Sárközy on their 70th birthdays
More informationABSTRACT. In this note, we find all the solutions of the Diophantine equation x k = y n, 1, y 1, k N, n INTRODUCTION
Florian Luca Instituto de Matemáticas UNAM, Campus Morelia Apartado Postal 27-3 (Xangari), C.P. 58089, Morelia, Michoacán, Mexico e-mail: fluca@matmor.unam.mx Alain Togbé Mathematics Department, Purdue
More informationDiophantine equations for second order recursive sequences of polynomials
Diophantine equations for second order recursive sequences of polynomials Andrej Dujella (Zagreb) and Robert F. Tichy (Graz) Abstract Let B be a nonzero integer. Let define the sequence of polynomials
More informationA Remark on Prime Divisors of Lengths of Sides of Heron Triangles
A Remark on Prime Divisors of Lengths of Sides of Heron Triangles István Gaál, István Járási, Florian Luca CONTENTS 1. Introduction 2. Proof of Theorem 1.1 3. Proof of Theorem 1.2 4. Proof of Theorem 1.3
More informationBALANCING NUMBERS : SOME IDENTITIES KABERI PARIDA. Master of Science in Mathematics. Dr. GOPAL KRISHNA PANDA
BALANCING NUMBERS : SOME IDENTITIES A report submitted by KABERI PARIDA Roll No: 1MA073 for the partial fulfilment for the award of the degree of Master of Science in Mathematics under the supervision
More informationEven perfect numbers among generalized Fibonacci sequences
Even perfect numbers among generalized Fibonacci sequences Diego Marques 1, Vinicius Vieira 2 Departamento de Matemática, Universidade de Brasília, Brasília, 70910-900, Brazil Abstract A positive integer
More informationAndrzej Schinzel 80: an outstanding scientific career
1 / 18 Andrzej Schinzel 80: an outstanding scientific career Kálmán Győry Short biography 2 / 18 Professor Andrzej Schinzel world-famous number theorist, old friend of Hungarian mathematicians. Born on
More informationDiophantine Equations
Diophantine Equations Michael E. Pohst Institut für Mathematik Technische Universität Berlin May 24, 2013 Mordell s equation y 2 = x 3 + κ is one of the classical diophantine equations. In his famous book
More informationJAN-HENDRIK EVERTSE AND KÁLMÁN GYŐRY
EFFECTIVE RESULTS FOR DIOPHANTINE EQUATIONS OVER FINITELY GENERATED DOMAINS: A SURVEY JAN-HENDRIK EVERTSE AND KÁLMÁN GYŐRY Abstract. We give a survey of our recent effective results on unit equations in
More informationarxiv: v1 [math.nt] 29 Dec 2017
arxiv:1712.10138v1 [math.nt] 29 Dec 2017 On the Diophantine equation F n F m = 2 a Zafer Şiar a and Refik Keskin b a Bingöl University, Mathematics Department, Bingöl/TURKEY b Sakarya University, Mathematics
More informationAlgebra II. A2.1.1 Recognize and graph various types of functions, including polynomial, rational, and algebraic functions.
Standard 1: Relations and Functions Students graph relations and functions and find zeros. They use function notation and combine functions by composition. They interpret functions in given situations.
More informationInteger Solutions of the Equation y 2 = Ax 4 +B
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.4 Integer Solutions of the Equation y 2 = Ax 4 +B Paraskevas K. Alvanos 1st Model and Experimental High School of Thessaloniki
More informationDiophantine Equations Concerning Linear Recurrences
Diophantine Equations Concerning Linear Recurrences PhD thesis László Szalay Lajos Kossuth University Debrecen, 1999. Contents 1 Introduction 1 2 Notation and basic results 6 2.1 Recurrence sequences.......................
More informationCURRICULUM MAP. Course/Subject: Honors Math I Grade: 10 Teacher: Davis. Month: September (19 instructional days)
Month: September (19 instructional days) Numbers, Number Systems and Number Relationships Standard 2.1.11.A: Use operations (e.g., opposite, reciprocal, absolute value, raising to a power, finding roots,
More informationOn a Question of Wintner Concerning the Sequence of Integers Composed of Primes from a Given Set
On a Question of Wintner Concerning the Sequence of Integers Composed of Primes from a Given Set by Jeongsoo Kim A thesis presented to the University of Waterloo in fulfillment of the thesis requirement
More informationTHE PROBLEM OF DIOPHANTUS AND DAVENPORT FOR GAUSSIAN INTEGERS. Andrej Dujella, Zagreb, Croatia
THE PROBLEM OF DIOPHANTUS AND DAVENPORT FOR GAUSSIAN INTEGERS Andrej Dujella, Zagreb, Croatia Abstract: A set of Gaussian integers is said to have the property D(z) if the product of its any two distinct
More informationBOUNDS FOR THE SIZE OF SETS WITH THE PROPERTY D(n) Andrej Dujella University of Zagreb, Croatia
GLASNIK MATMATIČKI Vol. 39(59(2004, 199 205 BOUNDS FOR TH SIZ OF STS WITH TH PROPRTY D(n Andrej Dujella University of Zagreb, Croatia Abstract. Let n be a nonzero integer and a 1 < a 2 < < a m ositive
More informationBéla Brindza ( )
Béla Brindza (1958 2003) Béla Brindza was born in 1958 in Csongrád, where he completed both primary and secondary school. His gift for mathematics became manifest in his high school years. He participated
More informationDecision Problems for Linear Recurrence Sequences
Decision Problems for Linear Recurrence Sequences Joël Ouaknine Department of Computer Science, Oxford University (Joint work with James Worrell and Matt Daws) RP 2012 Bordeaux, September 2012 Termination
More informationFIFTH ROOTS OF FIBONACCI FRACTIONS. Christopher P. French Grinnell College, Grinnell, IA (Submitted June 2004-Final Revision September 2004)
Christopher P. French Grinnell College, Grinnell, IA 0112 (Submitted June 2004-Final Revision September 2004) ABSTRACT We prove that when n is odd, the continued fraction expansion of Fn+ begins with a
More informationThis image cannot currently be displayed. Course Catalog. Algebra II Glynlyon, Inc.
This image cannot currently be displayed. Course Catalog Algebra II 2016 Glynlyon, Inc. Table of Contents COURSE OVERVIEW... 1 UNIT 1: SET, STRUCTURE, AND FUNCTION... 1 UNIT 2: NUMBERS, SENTENCES, AND
More informationBinomial Thue equations and power integral bases in pure quartic fields
arxiv:1810.00063v1 [math.nt] 27 Sep 2018 Binomial Thue equations and power integral bases in pure quartic fields István Gaál and László Remete University of Debrecen, Mathematical Institute H 4010 Debrecen
More informationON THE LIMIT POINTS OF THE FRACTIONAL PARTS OF POWERS OF PISOT NUMBERS
ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), 151 158 ON THE LIMIT POINTS OF THE FRACTIONAL PARTS OF POWERS OF PISOT NUMBERS ARTŪRAS DUBICKAS Abstract. We consider the sequence of fractional parts {ξα
More informationPerfect Powers With All Equal Digits But One
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.7 Perfect Powers With All Equal Digits But One Omar Kihel Department of Mathematics Brock University St. Catharines, Ontario L2S
More informationARITHMETIC PROGRESSION OF SQUARES AND SOLVABILITY OF THE DIOPHANTINE EQUATION 8x = y 2
International Conference in Number Theory and Applications 01 Department of Mathematics, Faculty of Science, Kasetsart University Speaker: G. K. Panda 1 ARITHMETIC PROGRESSION OF SQUARES AND SOLVABILITY
More informationScope and Sequence Mathematics Algebra 2 400
Scope and Sequence Mathematics Algebra 2 400 Description : Students will study real numbers, complex numbers, functions, exponents, logarithms, graphs, variation, systems of equations and inequalities,
More informationCurriculum Catalog
2017-2018 Curriculum Catalog 2017 Glynlyon, Inc. Table of Contents ALGEBRA II FUNDAMENTALS COURSE OVERVIEW... 1 UNIT 1: SET, STRUCTURE, AND FUNCTION... 1 UNIT 2: NUMBERS, SENTENCES, AND PROBLEMS... 1 UNIT
More informationRoot separation for irreducible integer polynomials
Root separation for irreducible integer polynomials Yann Bugeaud and Andrej Dujella 1 Introduction The height H(P ) of an integer polynomial P (x) is the maximum of the absolute values of its coefficients.
More informationFINITENESS RESULTS FOR F -DIOPHANTINE SETS
FINITENESS RESULTS FOR F -DIOPHANTINE SETS ATTILA BÉRCZES, ANDREJ DUJELLA, LAJOS HAJDU, AND SZABOLCS TENGELY Abstract. Diophantine sets, i.e. sets of positive integers A with the property that the product
More informationSquares in products with terms in an arithmetic progression
ACTA ARITHMETICA LXXXVI. (998) Squares in products with terms in an arithmetic progression by N. Saradha (Mumbai). Introduction. Let d, k 2, l 2, n, y be integers with gcd(n, d) =. Erdős [4] and Rigge
More informationPower series and Taylor series
Power series and Taylor series D. DeTurck University of Pennsylvania March 29, 2018 D. DeTurck Math 104 002 2018A: Series 1 / 42 Series First... a review of what we have done so far: 1 We examined series
More informationHomework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers.
Morgan County School District Re-3 A.P. Calculus August What is the language of algebra? Graphing real numbers. Comparing and ordering real numbers. Finding absolute value. September How do you solve one
More information