Download Applications of Fibonacci Numbers: Volume 4 Proceedings of by Peter G. Anderson (auth.), G. E. Bergum, A. N. Philippou, A. PDF

By Peter G. Anderson (auth.), G. E. Bergum, A. N. Philippou, A. F. Horadam (eds.)

This ebook includes thirty-three papers from one of the thirty-eight papers offered on the Fourth overseas convention on Fibonacci Numbers and Their functions which was once held at Wake woodland collage, Winston-Salem, North Carolina from July 30 to August three, 1990. those papers were chosen after a cautious overview via popular referees within the box, and so they diversity from straightforward quantity idea to chance and information. The Fibonacci numbers and recurrence relatives are their unifying bond. it truly is expected that this e-book, like its 3 predecessors, could be worthy to analyze staff and graduate scholars attracted to the Fibonacci numbers and their purposes. March 1, 1991 The Editors Gerald E. Bergum South Dakota country college Brookings, South Dakota, U. S. A. Alwyn F. Horadam college of recent England Armidale, N. S. W. , Australia Andreas N. Philippou Minister of schooling Ministry of schooling Nicosia, Cyprus xv THE ORGANIZING COMMITTEES neighborhood COMMITTEE foreign COMMITTEE Howard, Fred T. , Co-Chair Horadam, A. F. (Australia), Co-Chair Waddill, Marcellus E. , Co-Chair Philippou, A. N. (Cyprus), Co-Chair Hayashi, Elmer okay. Ando, S. (Japan) Bergum, G. E. (U. S. A. ) Vaughan, Theresa Harrell, Deborah Bicknell-Johnson, M. B. (U. S. A. ) Campbell, Colin (Scotland) Filipponi, Piero (Italy) Kiss, P. (Hungary) Turner, J. C. (New Zealand) xvii record OF participants TO THE convention *ALFORD, CECIL zero. , (coauthor Daniel C. Fielder) "Pascal's Triangle: best Gun or simply one of many Gang?" *ANDERSON, PETER G. , "A Fibonacci-Based Pseudo-Random quantity Generator.

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It is also known (see [7, Theorem 3]) that orc/( - b) Ip(a, b). (6) 3. PROOFS OF THE LEMMA AND THE MAIN THEOREM Proof of Lemma 1: Since ord( - b) I residues modulo p if ord( - b) =ord( - b/), it follows that - band - b' are both quadratic (p -1)/2 and are both quadratic non-residues modulo p if ord( - b) 1 (p - 1)/2. Thus, b'/b is a quadratic residue modulo p. Let t = ord( - b). It suffices to prove that rt ::: (mod p). (7) We have that r 2t ::: [( -b' )/( _b)]t::: 1 (mod p). Hence, if t is odd, then r t ::: _(_r)t::: ±1 (modp) and r t ::: 1 (mod p) by the definition of r.

1) f(x) = E~=o (2av+b)x tl ( av +b). 2) such that Theorem: Let P = 4am+3 be a prime. 3) 2 for n == - 2m - d (mod P) where d is an integer and d = b +3)/2a. Further, since b is odd, b = 2w+l makes 2(w 2 + w + 1) = ad. 2), we get Now, let n=uP - 2m - dj then C(2,n) is the coefficient of x uP in the expansion of x d + 2m (f(x))2. 4) where the sum extends over all nonnegative v, t satisfying v(av+b)+t(at+b)+2m+d=uP. The condition under the summation can be written, after multiplication by 4a, 4a(av+b)+4at(at+b)+8am+4ad=4auP, so that 4a(av+b)+4at(at+b)+8am+2b2+6=4auP, which leads to (2av+b)2 + (2at+b)2 = (4au-2)P.

Giordano A FEW BRIEF HISTORICAL NOTES ON PERFECT CUBES (1) In 1888, the first perfect magic cube ever constructed was of order 8, and was placed in "The Memoirs of the National Academy of Science" [3]. (2) Martin Gardner defines a perfect magic cube as follows: "A perfect magic cube is a cubical array of positive integers from 1 to N3 such that every straight line of N cells adds up to a constant. These lines include the orthogonals (the lines parallel to an edge), the two main diagonals of every orthogonal cross section and the four space diagonals.

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