[VA] SRC #016 - Pi Day 2024 Special

[Valentin Albillo]

      
Hi, all,

First of all, thank you very much to all of you who posted in this thread for your interest and appreciation, let alone your valuable contributions. A few comments before I post my original solutions. Let's see ,,.

Gerson W. Barbosa Wrote:I’ll take the opportunity to present my own polynomial equation of that kind:

x⁷ + 2 x⁶ + 3 x⁵ + 4 x⁴ + 3 x³ + 2 x² + x - 19100/3 = 0

I was expecting you to come with a fantastic input for Pi Day and true to form, you certanly delivered with that awesome "symmetric" equation, Gerson, which I'd never seen before and which I do like it, a lot.

This kind of equations can be reduced by using the change of variable x = z + 1/z, which almost halves the degree, so your equation would be reduced from 7^th-degree (usually unsolvable algebraically, thus requiring hyperelliptic functions) to 4^th-degree, which is algebraically solvable.

C.Ret Wrote:I am trying hard to solve this equation with my HP-15C, hoping that numeric integration, solver and hyperbolic trigonometric are of any use.

They certainly are and thanks for participating. I know that you like to be able to use physical calcs in my challenges (as opposed to ultra-fast virtual ones) so these 7 appearances are perfectly suited for you.

J-F Garnier Wrote:The most memorable unexpected appearance of pi for me was with the expression: sum(n=1 to Inf,1/n²) = π²/6. So simple ...

Well, come to that this is much simpler:

      

J-F Garnier Wrote:Here is my keystroke sequence on the 71B, with my Math 2B to save a few keystrokes:

Indeed, those INTEG, IX, FROOT, FX keywords really help, perhaps it even fits in just one line when you also omit the A variable (else you need a DESTROY ALL to guarantee A is not a matrix).

I've always hated those needlessly verbose INTEGRAL, FNROOT, IVAR and FVAR keywords, most especially when you've got a puny 22-char display. I wonder who was the inept that decided on those names, probably the same one that decided to waste 2.5 Kb in the useless abomination known as CALC mode or the unused 5 Kb in the original Math Pac, among other assorted imbecilities.

J-F Garnier Wrote:While experimenting a bit, I found a few interesting similar expressions:

Yes, they're nice alright but not that similar as they do not involve ϕ, the Golden Ratio, which is what I find appealing about this appearance (reversing the equation, \(\pi\), a mighty transcendental number, appears as a simple function of ϕ, the simplest irrational algebraic number and a modest, unassuming infinite product.)

J-F Garnier Wrote:I like these two ones!

Here we are with π approximations, not exact value given by infinite series.

Thank you and yes, among the 7 appearances dealt with here some are exact (in the limit) and others are prefixes of \(\pi\) (the N first digits for various N). For such simple equations, these two are very close approximations.

C.Ret Wrote:The next appearance is expected in no less than 243 hrs (10 days)

Also, 243 = 3⁵, twin primes !

jonakeys Wrote:edit: remove code panel since it's not allowed

Thanks for removing them, jonakeys, and also for including the keycodes in the nicely formatted program listings. Also, congratulations, you nailed the results.

ttw Wrote:All the funny Pi stuff connects with a circle, not necessarily directly from a circle, but maybe through angles.

O RLY ? Pray tell, where do you find the connection with circles or angles here, in this example I posted at the beginning of my OP ?

1 DESTROY ALL @ RANDOMIZE 260 @ FIX 4 @ INPUT K
2 N=0 @ FOR I=1 TO K @ N=N-MOD(IROUND(RND/RND),2) @ NEXT I @ DISP 1-4*N/K

>RUN

? 1E5 -> 3.1416

Or perhaps in the GCD-related Second appearance in my OP ? Do you think the GCD (Greatest Common Divisor) function has something to do with circles or angles ? Please explain how come.

DavidM Wrote:If the past is any predictor of the future, I'm sure that Valentin's version will be speedy i[...]

Well, not this time. I didn't intend for this to be a regular challenge but a nice celebration of \(\pi\). I just wanted interested people to write simple, minimal code or key sequences to get various incarnations of \(\pi\) to appear on their calc's displays, so no complicated programming, no clever tricks, no timings, just seeing \(\pi\) show up in unexpected places. Thus, this time my original solutions are no better and in particular no faster than those posted here, which are pretty good to begin with.

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I'll post my original solutions and comments next Thursday so you've got still a few days to try and post solutions to the remaining, yet-to-address appearances, namely the 4^th, 6^th and 7^th. Gotta Catch 'Em All !

V.
Edit: two typos.