Yet another Fibonacci mini-challenge (HP-42S/Free42)

[Gerson W. Barbosa]

(12-07-2018 06:31 PM)pier4r Wrote:  

(12-07-2018 03:41 PM)Gerson W. Barbosa Wrote:  While testing it, I noticed that \(\varphi^{39}\approx \frac{\pi ^{17}}{2}\). A little tweaking, making use of the apparent \(\sqrt{2}\) factors present in both sides of the expression, gives

How? I mean it is not that one computes pi to the 17 every other day. I am really interested how it comes to your mind to compute such values.

Let’s blame it on the RPN stack which makes for quick and easy calculations. For example, fill the stack with pi and keep pressing the × key for its successive powers. After 16 keypresses the number on the display is 282844564.3, which almost matches the digits of the square root of 8. Another example: fill the stack with pi again and press × + three times. Now you get 141.4265649. Again the first five digits look familiar, don’t they? That’s the sum of the first four powers of pi. Same when raising phi to the 39th power...

Edited to fix a mistake as pointed out by Valentin below.