By Heiko Harborth (auth.), A. N. Philippou, A. F. Horadam, G. E. Bergum (eds.)

This ebook includes nineteen papers from one of the twenty-five papers provided on the moment foreign convention on Fibonacci Numbers and Their functions. those papers were chosen after a cautious evaluate by means of popular referee's within the box, and so they variety from user-friendly quantity concept to likelihood and records. The Fibonacci numbers are their unifying bond. it truly is expected that this publication can be priceless to analyze employees and graduate scholars drawn to the Fibonacci numbers and their functions. October 1987 The Editors Gerald E. Bergum South Dakota nation college Brookings, South Dakota, U.S.A. Andreas N. Philippou college of Patras Patras, Greece Alwyn F. Horadam college of recent England Armidale, N.S.W., Australia xiii THE ORGANIZING COMMITTEES neighborhood COMMITTEE INTERN A TIONAL COMMITTEE Bergum, G., Chairman Philippou, A. (Greece), Chairman Edgar, H., Co-chalrman Horadam, A. (Australia), Co-chalrman Bergum, G. (U.s.A.) Thoro, D. Kiss, P. (Hungary) Johnson, M. lengthy, C. (U.S.A.) Lange, L.

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**Additional resources for Applications of Fibonacci Numbers: Proceedings of The Second International Conference on Fibonacci Numbers and Their Applications’ San Jose State University, California, U.S.A. August 1986**

**Example text**

Bumby [1]. } is weakly uniformly distributed mod m, if for every j relatively prime to m, lim N-_ 1 ~m ; a) - 0(m)' ~j, m ; a) (3) provided that there exist infinitely many n for which a .. is relatively prime to m. 0(,) is the Euler's tortient function and AN(j, m; a) is the same notation as in Definition 1 and B~m; a) denotes the number of indices n between 1 and N such that an is relatively prime to m. This definition is due to Narkiewicz [9]. Definition ~:. A sequence of integers a - {an} is uniformly distributed in (Z/mZ)*, if, for every invertible element j in Z I mZ , (4) if every an is relatively prime to the modulus m.

J-O AJ a. nJ [ a. (J .. +1)p - ~ L. r=O a r a. J is a zero of the polynomial (2), we may write (5) We now apply the multinomial theorem to the expression on the right of (5). The number of terms in this expansion depends on the number of non-zero coefficients ar • For notational simplicity, we will assume that each ar ~ O. This does not affect the generality of our results. •• , n.. such that no + ••. + n.. = p. We first prove the following result. • , n.. are non-negative integers such that no + .

Bull. Soc. Hath. France 6 (1878): pp 49-54. Kenji Nagasaka and Shiro Ando SYMMETRIC RECURSIVE SEQUENCES MOD M Distribution properties of integer sequences have been widely studied from various points of view. The sequence of Fibonacci numbers {Fn} is, of course, one of the main targets for this study. Indeed, {log F Il} is uniformly distributed mod I, so that {F,,} obeys Benford's law, detailed study of which is carried out in [6]. In this note we are going to treat uniform distribution properties of certain recursive integer sequences in residue classes.