Pi Formulas, Algorithms and Computations (2009)

> Every file that could possibly exist? > That's right! Every file you've ever created, or anyone else has created or will create! Copyright infringement? It's just a few digits of π! They were always there!

How about prior-art for Patents? Every invention is already written in Pi!

Honestly his output is insane in terms of both quality and quantity. There's a list of his projects on his homepage (https://bellard.org).

The basic idea with LLL is that it's a sort of generalized Euclid's algorithm. Euclid's algorithm can be made to find the smallest integer relation between two numbers that sums to zero (efficiently). You can generalize Euclid's algorithm to more numbers and to use vectors instead of base integers, except now the "divide, take the integer quotient and subtract" becomes "take off the integral component of the projection and subtract, then swap entries to try and make progress".

Lattice reduction techniques give a polynomial algorithm to factor polynomials with integral coefficients (does the polynomial factor into other polynomials with integral coefficients). To use lattice reduction techniques, you set up a "lattice basis" that has successive powers of pi (pi^0, pi^1, pi^2 etc) up to a certain accuracy and then tries to find an integer relation between them. Once one is found, it's checked to see if it actually equals pi by other techniques.

For an introduction to LLL, I've found Chee Yap's book to be good (see chapter VIII) [2].

Plouffe has a little bit of history about the discovery and how Bailey and Borwein muscled in [3].

[0] https://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%...

[1] https://en.wikipedia.org/wiki/Integer_relation_algorithm

[2] http://160592857366.free.fr/joe/ebooks/ShareData/Fundamental...

[3] http://xahlee.info/math/Simon_Plouffe_pi_formula.html

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3919892

The idea is you begin with an elementary integral that yields the underlying constant, such as pi, and then accelerate its convergence by using a polynomial.

Ramanujan-like pi formulas can be derived similar to the Fabrice Bellard one, but with an inverse cosine instead of elliptic integrals. Visually, the formulas look like the Ramanujan pi formulas, but the derivation is elementary and it computes pi, not the reciprocal of pi. The Bellard and Simon formulas use inverse trig functions as well. I think he uses arctan.

If the position is below 5 trillion, you might want to calculate them once with Bellard's software and use that data.

E: if you want to have reliable, consistent results that is