Exploring the Absence of Certain Numbers in the Fibonacci Sequence Modulo 11

The Fibonacci sequence is a fascinating mathematical object with numerous applications in both pure and applied mathematics. In this article, we delve into why certain numbers cannot appear in the Fibonacci sequence when considered modulo 11. We will use two methods to prove this: one based on Binet's formula and another using the periodicity of the sequence.

Method Using Binet's Formula

Binet's formula for the nth Fibonacci number is given by:

Working modulo 11, we note that (sqrt{5} equiv pm 4 pmod{11}) because ((pm 4)^2 equiv 5 pmod{11}). Using the positive square root, we find:

Since 11 is prime, by Fermat's Little Theorem, (8^{11-1} equiv 4^{11-1} equiv 1 pmod{11}). Moreover, (4^5 equiv 1 pmod{11}) and (8^5 equiv 1 pmod{11}). This implies that there are at most ten distinct values for (F_n pmod{11}). We can compute the Fibonacci numbers modulo 11 for (n 0, 1, 2, ldots, 9) and find that the only possible values are (0, 1, 2, 3, 5, 8, 10). Therefore:

Method Using Periodicity

Consider the terms of the Fibonacci sequence modulo 11: 11, 2, 3, 5, 8, 2, 10, 0, 10, 1, 1. This sequence is periodic with a period of 10. In other words, the sequence repeats every 10 terms. The sequence includes the numbers 0, 1, 2, 3, 5, 8, and 10, but no 4, 6, 7, or 9. Therefore:

Since the sequence modulo 11 does not include 4, 6, 7, or 9, no Fibonacci number can be of the form 11k4, 11k6, 11k7, or 11k9. This completes the proof.

Conclusion

In summary, the absence of certain numbers in the Fibonacci sequence when considered modulo 11 is a result of both the periodicity of the sequence and its unique properties derived from Binet's formula. This periodicity ensures that the sequence modulo 11 is limited to a set of specific values, which does not include 4, 6, 7, or 9. These insights provide a deeper understanding of the intrinsic structure of the Fibonacci sequence and its behavior under modular arithmetic.