# The 1 millionth fibonacci number

time echo 'n=1000000;m=(n+1)/2;a=0;b=1;i=0;while(m){e[i++]=m%2;m/=2};while(i--){c=a*a;a=c+2*a*b;b=c+b*b;if(e[i]){t=a;a+=b;b=t}};if(n%2)a*a+b*b;if(!n%2)a*(a+2*b)' | bc
EDIT: Trolling crap removed ;) takes approx 6 secs on a Core 2 Duo @ 2GHz, and 15 secs on atom based netbooks! uses monoid (a,b).(x,y)=(ax+bx+ay,ax+by) with identity (0,1), and recursion relations: F(2n-1)=Fn*Fn+F(n-1)*F(n-1) F(2n)=(Fn+2*F(n-1))*Fn then apply fast exponentiation to (1,0)^n = (Fn,F(n-1)) . Note that: (1,0)^-1=(1,-1) so (a,b).(1,0) = (a+b,a) and (a,b)/(1,0)=(a,b).(1,0)^-1=(b,a-b) So we can also use a NAF representation to do the exponentiation,http://en.wikipedia.org/wiki/Non-adjacent_form , it's also very fast (about the same, depends on n): `time echo 'n=1000000;m=(n+1)/2;a=0;b=1;i=0;while(m>0){z=0;if(m%2)z=2-(m%4);m=(m-z)/2;e[i++]=z};while(i--){c=a*a;a=c+2*a*b;b=c+b*b;if(e[i]>0){t=a;a+=b;b=t};if(e[i]<0){t=a;a=b;b=t-b}};if(n%2)a*a+b*b;if(!n%2)a*(a+2*b)' | bc`
Sample Output
```19532821287077577316320149475962563324435429965918733969534051945716\
...
54385552378378141386675079286837205802043347225419033684684301719893\
411568996526838242546875

(over two hundred thousand digits, too long to display fully!)

real	0m6.614s
user	0m5.888s
sys	0m0.024s

```

-135
2009-09-10 09:00:44

## 1 Alternatives + Submit Alt

• Calculates nth Fibonacci number for all n>=0, (much faster than matrix power algorithm from http://everything2.com/title/Compute+Fibonacci+numbers+FAST%2521 ) n=70332 is the biggest value at http://bigprimes.net/archive/fibonacci/ (corresponds to n=70331 there), this calculates it in less than a second, even on a netbook. UPDATE: Now even faster! Uses recurrence relation for F(2n), see http://en.wikipedia.org/wiki/Fibonacci_number#Matrix_form n is now adjusted to match Fn at wikipedia, so bigprimes.net table is offset by 1. UPDATE2: Probably fastest possible now ;), uses a simple monoid operation: uses monoid (a,b).(x,y)=(ax+bx+ay,ax+by) with identity (0,1), and recursion relations: F(2n-1)=Fn*Fn+F(n-1)*F(n-1) F(2n)=Fn*(2*F(n-1)+Fn) then apply fast exponentiation to (1,0)^n = (Fn,F(n-1)) . Note that: (1,0)^-1=(1,-1) so (a,b).(1,0) = (a+b,a) and (a,b)/(1,0)=(a,b).(1,0)^-1=(b,a-b) So we can also use a NAF representation to do the exponentiation,http://en.wikipedia.org/wiki/Non-adjacent_form , it's also very fast (about the same, depends on n): `time echo 'n=70332;m=(n+1)/2;a=0;b=1;i=0;while(m>0){z=0;if(m%2)z=2-(m%4);m=(m-z)/2;e[i++]=z};while(i--){c=a*a;a=c+2*a*b;b=c+b*b;if(e[i]>0){t=a;a+=b;b=t};if(e[i]<0){t=a;a=b;b=t-b}};if(n%2)a*a+b*b;if(!n%2)a*(a+2*b)' | bc` Show Sample Output

-136
time echo 'n=70332;m=(n+1)/2;a=0;b=1;i=0;while(m){e[i++]=m%2;m/=2};while(i--){c=a*a;a=c+2*a*b;b=c+b*b;if(e[i]){t=a;a+=b;b=t}};if(n%2)a*a+b*b;if(!n%2)a*(a+2*b)' | bc
· 2009-09-10 08:58:47

### What Others Think

My C version is faster. Stop posting duplicates. hank · 632 weeks and 1 day ago
Stop it Escher. You're just spamming now. Quit posting this and more variants. We get it already! linuxrawkstar · 632 weeks and 1 day ago
Your C version is slower,in fact ~47secs vs my ~36secson a Core 2 Duo. I have put a message for people like you in capitals in the description section. No go away little buzzing insect. Escher · 632 weeks and 1 day ago
Don't be too hard on this MrMerry/Escher degenerate... Desperate attempts to get some attention on the internet is his only way of communicating with the world outside his mama's basement... sklm · 632 weeks and 1 day ago
Please only post comments if you have required skillz to compete against me. I know it's depressing for you to witness true greatness and realise how insignificant you are, but that's life sklm, you always were and always will be an irrelevant ant. Escher · 632 weeks and 1 day ago
"Please only post comments if you have required skillz to compete against me. I know it's depressing for you to witness true greatness and realise how insignificant you are, but that's life sklm, you always were and always will be an irrelevant ant." LMAO!!! sklm · 632 weeks and 1 day ago
Well now. Someone has a voting script... SuperFly · 632 weeks and 1 day ago
can this be made faster by using multiple CPUs ? alperyilmaz · 632 weeks ago
Not really, though it can be made faster by using the recurrence relations from the wikipedia article more efficiently. Check for an update soon :) For the record, mathematica uses an algorithm which is about 4 times faster. Escher · 632 weeks ago
I don't think this was a voting script; I'd like to think this is a reward for arrogance. Escher: a truly great person is humble. What bothers you is that the reverse is not necessarily true. And as long as that bothers you, you will be what you are, and will not grow further. sitaram · 632 weeks ago
Don't be so arrogant, dude ! It's just a shell command... I don't care for these numbers in any shell script... useless for me, sorry. alvinx · 631 weeks and 5 days ago

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