Goldbach conjecture - like statement for the Fibonacci sequence

The strong Goldbach conjecture states that every even integer greater than two is the sum of two primes. The weak Goldbach conjecture is very similar, instead modifying the sum from two primes to three primes.

The prime numbers are on example of a series, as is the Fibonacci series. Extending the Goldbach conjecture idea to the Fibonacci series, I am wondering about the status of this statement:

For every (even/odd) integer greater than K, it can be rewritten as the sum of N Fibonacci numbers (N and K are constant)

Extra: the Fibonacci series is simply one type of series that can be extended to the Goldbach conjecture idea. Are there any other special series that are true or false to this idea?

I am trying to explore the mathematics behind this baffling conjecture. All help is appreciated, Thank you.

1 Answer

By Zeckendorf's theorem every natural number $n\geq 1$ can be written in a unique way as a sum of Fibonacci numbers with indices $\geq 2$ and non-consecutive. In particular the numbers of the form $F_2+F_4+\ldots+F_{2M}$ immediately disprove your conjecture. Also: Fibonacci numbers have an exponential growth, hence they cannot be an additive base of finite order.

• I agree with you that the exponential growth of the Fibonacci sequence is enough to disprove the conjecture, but I think your remark about the Zeckendorff representation is irrelevant. The mere existence of a long Zeckendorff representation isn't enough to "immediately" rule out the short representation that OP wants. – MJD Apr 10 '17 at 16:51
• @MJD: the point is that the Zeckendorf representation is the most compact (i.e. with fewer terms) among the representations as sums of Fibonacci numbers. So if the Zeckendorf representation is not able to represent every (large enough) number in terms of a finite number of terms $F_k$, the same holds for every other representation. – Jack D'Aurizio Apr 10 '17 at 16:53
• – Jack D'Aurizio Apr 10 '17 at 16:54