## Fibonacci Conjecture

### January 23, 2015

We test primality using the Miller-Rabin algorithm:

`from random import randint`

```
```

`def isPrime(n, k=5): # miller-rabin`

if n < 2: return False

for p in [2,3,5,7,11,13,17,19,23,29]:

if n % p == 0: return n == p

s, d = 0, n-1

while d % 2 == 0:

s, d = s+1, d/2

for i in range(k):

x = pow(randint(2, n-1), d, n)

if x == 1 or x == n-1: continue

for r in range(1, s):

x = (x * x) % n

if x == 1: return False

if x == n-1: break

else: return False

return True

Then we generate the fibonacci sequence item by item, testing each member of the sequence for the conjecture:

`fminus2, fminus1, f, n = 0, 1, 1, 1`

```
```while not isPrime(f*f+41):

fminus2 = fminus1

fminus1 = f

f = fminus2 + fminus1

n = n + 1

`print n`

It takes a while, but when *n* = 12586 the conjecture is disproved; *F _{n}* =

2437420528658316261009349390793966957742064\

3628861129205119104129367296544863664132318\

7350629550946523694153755521722137479950927\

2904723884962366758282082558958533077607016\

4703846370983386287796589758104627824352090\

5662005007112216576623475679256710071871042\

3501925305429172597994769639007117036154468\

7180679967509632642871862665093652981386467\

8910317854243133922877435406205076602327076\

9741103525885355233426517035556242651050139\

7776152157485442589142262270805545603989018\

2738554122228622158431371645352398507835723\

0385583847730548224182907279173376294450080\

9559053247797188570862253302394816309497146\

6878325147926036684128681457928096551878450\

9048663514751789277765677734737548105702983\

9734854764440251783035033215144047874666733\

6848868926251340058764754077781140566043315\

8701933074493040543009488664637360050636904\

2928079603209789838364413285025609380384265\

6795053352943252616137226071900435738575737\

1979845493254607828392109996976472074994766\

9033598594913864908202612667104449679242977\

8610516507362889175957975232663636222152077\

4284526382848518239624979685302202912236641\

6943769232769303904917019647139581084704832\

7272516266180237248581730637609553420348612\

0484912018032012452608038735409013648590994\

8240242575154900347260104409588725662982761\

8930795229966774742603711212498750506258076\

3014931449402618392241302765309360591836332\

5458917719554093037261461679173796989091018\

2435816239415372148826244578310525764913283\

5242405725134042227992345138785534218377273\

5053838856017355194609598851478158802143512\

7175941347773542411307602896674709526451628\

6476071607991857431689522947614987960169330\

6722813421495472939532252545388730790498301\

2251572189695241349427758764490786113582769\

8645909666431556968864724496839492828557311\

9170002013318196219345226290957660638411642\

0016581274984591009084728165483905358516182\

0297135028779916414354206794185844039364018\

8608957647160096027031880341964822877335354\

5751715205632025400536873014390149732416863\

9085890424786551710413554658241864854000794\

6745670650479244297452105370763155766676763\

7318438261153992762039335049099708136846624\

8277001806713678125062905001779238666279561\

6632647722206547535282413463861123144939517\

6474406575548095847232978261553672514077110\

1662200616963869051516130934571518303860888\

9635879329162191473644658218460874561795754\

7161483044610603455086191314143678151570447\

9169448264328530870952452811943101857223733\

6743709285893930610721261923392743649444251\

2593507974839519719984251378323935971154303\

0401762641878184079758301800088653799694925\

1691510246396397638712811252669104365214937\

6779473254356292498767183110896869322621345\

2857548409354754231404289782181584228127633\

62455056

I’m writing in Python instead of Scheme because I happened to have a Python window open, with an `isPrime`

function already loaded, when I saw the conjecture. That takes a while to run because the numbers quickly become large.

This is a good example of a conjecture that seems to be true; it takes a long time to find a counter-example (with 2630 digits!) so it fools you into thinking that it might be correct. There are other famous conjectures — the Collatz conjecture, the twin-primes conjecture, the conjecture that all perfect numbers are even, the Riemann hypothesis — for which there is strong empirical evidence but no proof.

You can run the program at http://programmingpraxis.codepad.org/Q4e8Xt8V.

Quick and dirty brute-force method in python. Copy and pasted the isprime function, made a fibb generator, and started testing for anomalies.

Detected 2^2 + 41 as the odd one out:

NEVER MIND, IGNORE MY PREVIOUS COMMENT, NOT THE SOLUTION.

I am truly a poop.

Fails for F_0, as 41 is prime.

I think that according to the definition in the question the value of n in the model solution is off by 2.

For that definition, the initialisation line would be

i took 1 as first fibonacci number and run my code in several ranges as my machine lets me.

it seems conjecture is true. though if it is not, i will not be surprised.

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