Fibonacci Conjecture

January 23, 2015

The sequence of Fibonacci numbers is defined as F0 = 0, F1 = 1, Fn = Fn−2 + Fn−1. It has been conjectured that for any Fibonacci number F, F2 + 41 is composite.

Your task is to either prove the conjecture to be true or find a counter-example that demonstrates it is false (hint: this is not a blog about proving math theorems). When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


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7 Responses to “Fibonacci Conjecture”

  1. Brett Warren said

    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:

    def isprime(number):
    	from math import ceil
    	if number <= 1:
    		return False
    	elif number == 2:
    		return True
    	elif number % 2:
    		return False
    		for n in range(3, number**0.5, 2):
    			if not number % n:
    				return True
    		return False
    def fibb_gen():
    	(n, m) = (1, 1)
    	while True:
    		yield (n + m)
    		(n, m) = (m, (n + m))
    if __name__ == "__main__":
    	a = fibb_gen()
    	for n in a:
    		if n % 2:
    		if not isprime(n**2 + 41):
    			print(n, n**2 + 41)
  2. Brett Warren said


    I am truly a poop.

  3. sbmassey said

    Fails for F_0, as 41 is prime.

  4. Jan Van lent said

    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

    fminus2, fminus1, f, n = 0, 1, 1, 2
  5. ironiya said

    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.

    import math
    def is_perfect_square(x):
    	return int(math.sqrt(x)) == math.sqrt(x)
    def is_fibo(x):
    	c = 5*x*x
    	return is_perfect_square(c+4) or is_perfect_square(c-4)
    def is_prime(x):
    	c = int(math.sqrt(x))+2
    	if (x%2==0 and x!=2) or is_perfect_square(x): return False
    		for i in range(2, c):
    			if x%i==0: return False
    	return True
    def minus_four(fib):
    	if is_fibo(fib) and fib>0: return is_perfect_square(5*fib*fib-4)
    def fibo(x):
    	if x<2: return 1
    	else: return fibo(x-1)+fibo(x-2)
    def fibonacci_conjecture(end):
    	return (True not in map(is_prime, [fibo(i)*fibo(i)+41 for i in range(end+1)]))
    def second_way(end):
    	return (True not in map(is_prime, [i**2+41 for i in range(end+1) if is_fibo(i)]))
    def third_way(end):
    	return (True not in map(is_prime, [fibo(i)*fibo(i)+41 for i in range(end+1) if minus_four(i)]))

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