PROBLEMS from volume 33 (1995) issue #1 of The Fibonacci Quarterly.

Copyright ◯^{c} 1995 by The Fibonacci Association.

                         Inquiries to: Fibonacci@MathPro.com

ELEMENTARY PROBLEMS AND SOLUTIONS Edited by Stanley Rabinowitz

Please send all material for ELEMENTARY PROBLEMS AND SOLUTIONS to Dr. STAN-LEY RABINOWITZ; 12 VINE BROOK RD; WESTFORD, MA 01886-4212 USA. Correspondence may also be sent to the problem editor by electronic mail to Fibonacci@MathPro.com on Internet. All correspondence will be acknowledged.

Each solution should be on a separate sheet (or sheets) and must be received within six months of publication of the problem. Solutions typed in the format used by The Fibonacci Quarterly will be given preference. Proposers of problems should normally include solutions. Proposers should inform us of the history of the problem, if it is not original. A problem should not be submitted elsewhere while it is under consideration for publication in this column.

BASIC FORMULAS

The Fibonacci numbers F_{n} and the Lucas numbers L_{n} satisfy

F_{n+2} = F_{n+1} + F_{n}, F_{0} = 0, F_{1} = 1;
L_{n+2} = L_{n+1} + L_{n}, L_{0} = 2, L_{1} = 1.

Also, α = (1 + \sqrt{5})/2, β = (1 − \sqrt{5})/2, F_{n} = (α^{n} − β^{n})/\sqrt{5}, and L_{n} = α^{n} + β^{n}.

PROBLEMS PROPOSED IN THIS ISSUE

\underline{B–778} Proposed by Eliot Jacobson, Ohio University, Athens, Ohio

While paging through an old text in our library, I found a tattered and yellowed page, clearly out of place, as if it had been torn from a book, long forgotten. After months of tedious work, I have completed the translation of the scribbled markings on that page. In the margin was noted:

“I have found a truly wondrous demonstration of the following theorem, unfortunately the margin of this page is too small to contain it.”

And then followed: Fibonacci’s Last Theorem: The equation x^{n} + y^{n} = z^{n} has no non-trivial solutions consisting entirely of Fibonacci numbers, for n ≥ 2.

Can you supply the missing proof?

\underline{B–779} Proposed by Andrew Cusumano, Great Neck, New York Find integers a, b, c, and d (with 1 < a < b < c < d) that make the following an identity:

F_{n} = F_{n−a} + 6F_{n−b} + F_{n−c} + F_{n−d}.

\underline{B–780} Proposed by Zdravko F. Starc, Vršac, Yugoslavia Prove that

(a)  F_{1} · F_{2} · F_{3} ⋯ F_{n} ≤ exp(F_{n+2} − n − 1);
(b)  F_{1} · F_{3} · F_{5} ⋯ F_{2n−1} ≤ exp(F_{2n} − n);
(c)  F_{2} · F_{4} · F_{6} ⋯ F_{2n} ≤ exp(F_{2n+1} − n − 1).

\underline{B–781} Proposed by H.-J. Seiffert, Berlin, Germany

Let F (j) = F_{j}. Find a closed form for ∑_{k=0}^{n} F (k − ⌊\sqrt{k}⌋^{2}).

(The notation ⌊x⌋ denotes the greatest integer less than or equal to x.)

\underline{B–782} Proposed by László Cseh, Stuttgart, Germany and Imre Merény, Budapest, Hungary

Express (F_{n+h}^{2} + F_{n}^{2} + F_{h}^{2})(F_{n+h+k}^{2} + F_{n+k}^{2} + F_{k}^{2}) as the sum of three squares.

\underline{B–783} Proposed by David Zeitlin, Minneapolis, Minnesota Find a rational function P (x, y) such that

P (F_{n}, F_{2n}) = \frac{105n^{5} − 1365n^{3} + 1764n}{25n^{6} + 175n^{4} − 5600n^{2} + 5904}

for n = 0, 1, 2, 3, 4, 5, 6.