Also, since ∀ n > k : c

n

> 0 . F rom there we get that ∀ n > k : q

n

> q

k

. Whic h in turn results in:

| η − η

n

| ≤









Q

n

i =1

c

i

q

n +1

q

n









≤









Q

n

i =1

c

i

q

2

k









≈ κ

1



β

d

b

α

2

d

a



n − k



n !

k !



d

b

− 2 d

a

= κ

2



β

d

b

α

2

d

a



n − k

( n !)

2 d

a

− d

b

Since

exp ( n )

n !

is decreasing sup er-exponentially , the desired result is obtained.

C.3.3 Sub-Exp onen tial

The case satisfying the determinant constraint 4

β

d

b

> − α

2

d

a

can b e seen as a limit of the exp onen tial

con vergence case with

c → ∞

, therefore the derived con vergence is sub-exp onen tial. W e b eliev e this

sub-exp onen tial conv ergence to be p olynomial.

C.3.4 T ail Estimation

In the case of an exp onen tially conv erging GCFP , we found that from some p oin t the tail is approximately:

1 +

c

1 +

c

1+ ...

W e calculated the conv ergence v alue of 1-perio dic GCF s like this earlier. Therefore, we can improv e a

GCFP calculation b y substituting this tail at the ﬁnal step. The accuracy impro vemen t wasn’t analyzed,

but empiric results display an improv ement of a ﬁxed num b er of digits (for any large

n

). This in turn

allo wed us to improv e the complexit y of the MITM-RF algorithm.

D Collab orativ e Algorithm-Enhanced Mathematics

The Raman ujan Mac hine in its most general sense can be seen as a methodology to generate conjectures on

fundamen tal constan ts. The more computational pow er and the more time the algorithm runs on a selected

space of parameters, the more conjectures it may generate. Moreov er, since the Ramanujan Machine

pro duces conjectures on fundamental constants but not their pro ofs, w e realize that computational pow er

as well as proving p ow er (i.e. time sp ent b y an in telligent b eing trying to prov e or refute a conjecture) are

k ey assets for making the Ramanujan Machine more proliﬁc. It is the goal of this section to discuss how

one may leverage these facts ab out the Ramanujan Machine metho dology to inspire the wider comm unity

ab out mathematics and num b er theory .

W e created the Raman ujan Machine as an open source pro ject that is fully a v ailable to the communit y

on

www.RamanujanMachine.com

. So on, with our ongoing developmen t, individuals around the w orld

w ould b e able to donate their computational p ow er to the mission of discov ering new mathematical

structures and mathematical equations by downloading the Ramanujan Machine screen sav er on the

w ebsite. Similarly to SETI (Search for Extraterrestrial Intelligence), w e plan to hav e the Ramanujan

Mac hine screen sa ver distribute via BOINC the v arious computational tasks to every computer in the

net work, so when a computer is idle, the Ramanujan Machine is initiated.

W e b elieve this metho dology can inspire the greater communit y ab out mathematics. In order to

ac hieve this goal, the site

www.RamanujanMachine.com

includes an up-to-date record of some (and in a

short time ev ery) conjecture generated by the mac hine. When a sp eciﬁc computer in the netw ork discov ers

a new conjecture, the o wner of the laptop will receiv e the credit for contributing his or her computer

p o wer to discov er the conjecture and the credit is maintained in a leadership b oard. Since the Ramanujan

Mac hine is a conjecture-generating mac hine (similarly to m uch of the w ork of Raman ujan himself ), we let

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