MathDB

Given integer

n > 1

and real number

t \geq 1

P

is a parallelogram with four vertices

(0, 0), (0, t), (tF_{2n+1}, tF_{2n}), (tF_{2n+1}, tF_{2n} + t)

. Here,

{F_n}

is the

n

-th term of Fibonacci sequence defined by

F_0 = 0, F_1 = 1

and

F_{m+1} = F_m + F_{m-1}

. Let

L

be the number of integral points (whose coordinates are integers) interior to

P

, and

M

be the area of

P

, which is

t^2F_{2n+1}.

i) Prove that for any integral point

(a, b)

, there exists a unique pair of integers

(j, k)

such that

j(F_{n+1}, F_n) + k(F_n, F_{n-1}) = (a, b)

, that is,

jF_{n+1} + kF_n = a

and

jF_n + kF_{n-1} = b.

ii) Using i) or not, prove that

|\sqrt L-\sqrt M| \leq \sqrt 2.

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