Saturday, 16 February 2013

SATURDAY, 16 FEBRUARY 2013

Today is $47^{th}$ day of the year.

47 is a prime number.

47 is the $8^{th}$ Lucas Number, see http://oeis.org/A000032

The Lucas Number are like the Fibonacci numbers in that the $n^{th}$ number is the sum of the previous two numbers, i.e. the $(n-1)^{th}$ and the  $(n-2)^{th}$. In the case of the Fibonacci numbers the first two numbers are 0 and 1 giving the sequence
0, 1, 1, 2, 3,  5,  8, 13, 21, 34,  55,  89, 144, 233, 377, ...
whereas the Lucas Numbers start with 2 and 1 giving the following sequence:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, ...
You may be able spot an interesting link between these two series; the $n^{th}$ Lucas Number is equal to the sum of the  $(n-1)^{th}$ and the  $(n+1)^{th}$ Fibonacci Number i.e the eighth Lucas Number, 47 , is equal to the seventh Fibonacci Number, 13, plus the ninth Fibonacci Number, 34.
We could write this as:
$$L_{n}=F_{n-1}+F_{n+1}$$
Using the same nomenclature, Wikipedia informs us that there are a number of other identities:
\begin{align}L_{m+n} = L_{m+1}F_{n}+L_mF_{n-1}\end{align}
\begin{align}L_n^2 = 5 F_n^2 + 4 (-1)^n\end{align}
\begin{align}F_{2n} = L_n F_n\end{align}
\begin{align}F_n = {L_{n-1}+L_{n+1} \over 5}\end{align}
Equation 1
With $m=3$ and $n=5$ then $L_8 = L_4.F_5 + L_3.F_4 = 7.5 + 4.3 = 35 + 12 = 47$

Equation 2
With $n = 8$ then $L_8^2 = 5.F_8^2 + 4.(-1)^8 = 5.{21}^2 + 4 = 5.441 + 4 = 2,209 = 47^2$

Equation 3
With $n = 8$ then $F_{16} = L_8.F_8 = 47.21 = 987$

Equation 4
With $n = 7$ then $F_7 = {L_6 + L_8 \over 5} = {18 + 47 \over 5} = {65 \over 5} = 13$

1 comment:

Max said...

You make my brain boggle and wish I'd listened more in Maths.