### otajonov_sherzod's blog

By otajonov_sherzod, history, 5 weeks ago, Hey guys,

So there is this problem on acmp.ru that asks you to determine if a given number N is a fibonacci number where $1<= N <= 10^{100000}$.

Time: 8 sec. Memory: 16 MB

I tried to solve this with this bigint in c++ using brute force approach. But it gave TLE on test 14. Then I switched to python, did the same thing, and got AC. After some googling, I also found that a number N is a fibonacci number if $5*N^2+4$ or $5*N^2-4$ is a perfect square. So i tried it with python but it got WA on test 3, probably because of some precision errors.

Here are my attempts:

### Attempt 1 (TLE on test 14)


#include "bits/stdc++.h"
using namespace std;

int main(){
string r = "", line;
ifstream inp;
ofstream outp;
inp.open ("INPUT.txt");
outp.open ("OUTPUT.txt");
getline (inp, line);
bigint x = line, a = 0, b = 1, tmp;
while(a < x){
tmp = b; b = a+b; a = tmp;
}
if(a == x) r = "YES\n";
else r = "NO\n";

outp << r;
inp.close(); outp.close();
}



I didnt include some stuff like bigint struct, because code would be like 400/500 lines long with it.

### Attempt 2(AC)


from math import sqrt
with open("INPUT.TXT") as inp:
r='';
x=int(next(inp));
a, b = 0, 1;
while(a < x):
a, b = b, a + b;
if(a == x): r = 'YES';
else: r = 'NO';
with open("OUTPUT.TXT", 'w') as out:
out.write(r);



### Attempt 3( WA on test 3)


from math import sqrt

with open("INPUT.TXT") as inp:
r = '';
x = int(next(inp));
n = 5*x**2; sq1 = int(sqrt(n+4)); sq2 = int(sqrt(n-4));
if(sq1*sq1 == n+4 or sq2*sq2 == n-4): r = "YES";
else: r = "NO";
with open("OUTPUT.TXT", 'w') as out:
out.write(r);



So, I'm wondering how we could get AC with c++ for this problem, and if its possible to fix the precision error of this second approach.  Comments (7)
 » Auto comment: topic has been updated by otajonov_sherzod (previous revision, new revision, compare).
 » In the third solution, checking around 2 neighbouring numbers of the root would do.
 » Standard sqrt might produce incorrect values, so you can use additional binary searches instead.
 » There is one pretty easy and beautiful way to solve this problem (check whether number $N$ is perfect square) without using sqrt functions (which are not precise). Pick some random number $M$ and check whether $N$ gives correct quadratic residue modulo $M$. If no, $N$ can't be perfect square. Repeat that many times. If no contradiction was found, $N$ will be (with very high probability) perfect square. In that very problem, checking all $M$ in range $[3, 20]$ is enough to get accepted. code.
 » Why does attempt 2 give you AC and not TLE?
•  » » The time limit is 8 seconds which is very large so maybe thats why.
 » 5 weeks ago, # | ← Rev. 2 →   Hello hashing?There is an easier way to solve similar problems. Calculate all the fibonacci numbers from 0 to 106 (less is enough for this problem) and store them modulo M in a map/set. Then for any input number X, just check if X % M is in the map/set. M can be any large prime for the sake of simplicity.Works for fibonacci, factorial, polynomials and what not.