PROBLEMS from volume 30 (1992) issue #4 of The Fibonacci Quarterly.

Copyright ◯^{c} 1992 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 below 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–724} Proposed by Larry Taylor, Rego Park, NY

Dedicated to Dr. A. P. Hillman Let n be a positive integer. Prove that the numbers L_{n−1}L_{n+1}, 5F_{n}^{2}, L_{3n}/L_{n}, L_{2n},

F_{3n}/F_{n}, L_{n}^{2}, 5F_{n−1}F_{n+1} are in arithmetic progression and find the common difference.

\underline{B–725} Proposed by Russell Jay Hendel, Patchogue, NY and Herta T. Freitag, Roanoke, VA

Dedicated to Dr. A. P. Hillman (a) Find an infinite set of right triangles each of which has a hypotenuse whose length is a

Fibonacci number and an area that is the product of four Fibonacci numbers.

(b) Find an infinite set of right triangles each of which has a hypotenuse whose length is the product of two Fibonacci numbers and an area that is the product of four Lucas numbers.

\underline{B–726} Proposed by Florentin Smarandache, Phoenix, AZ

Dedicated to Dr. A. P. Hillman

Let d_{n} = P_{n+1} − P_{n}, n = 1, 2, 3, …, where P_{n} is the nth prime. Does the series

∑_{n=1}^{∞} \frac{1}{d_{n}}

converge?

\underline{B–727} Proposed by Ioan Sadoveanu, Ellensburg, WA

Dedicated to Dr. A. P. Hillman

Find the general term of the sequence (a_{n}) defined by the recurrence

a_{n+2} = \frac{a_{n+1} + a_{n}}{1 + a_{n+1}a_{n}}

with initial values a_{0} = 0 and a_{1} = (e^{2} − 1)/(e^{2} + 1), where e is the base of natural logarithms.

\underline{B–728} Proposed by Leonard A. G. Dresel, Reading, England

Dedicated to Dr. A. P. Hillman

If p > 5 is a prime and n is an even integer, prove that

(a) if L_{n} ≡ 2 (mod p), then L_{n} ≡ 2 (mod p^{2});

(b) if L_{n} ≡ −2 (mod p), then L_{n} ≡ −2 (mod p^{2}).

\underline{B–729} Proposed by Lawrence Somer, Catholic University of America, Washington, DC

Dedicated to Dr. A. P. Hillman

Let (H_{n}) denote the second-order linear recurrence defined by

H_{n+2} = aH_{n+1} + bH_{n},

where H_{0} = 0, H_{1} = 1, and a and b are integers. Let p be a prime such that p ∤ b. Let k be the least positive integer such that H_{k} ≡ 0 (mod p). (It is well-known that k exists.) If H_{n} ≢ 0 (mod p), let R_{n} ≡ H_{n+1}H_{n}^{−1} (mod p).

(a) Show that R_{n} + R_{k−n} ≡ a (mod p) for 1 ≤ n ≤ k − 1.

(b) Show that R_{n}R_{k−n−1} ≡ −b (mod p) for 1 ≤ n ≤ k − 2.