The Mystery of Fibonacci Partial Sums Divisible by 11: A Discovery

The Fibonacci sequence is a remarkable series where each number is the sum of the two preceding ones: (F_0 0), (F_1 1), and (F_n F_{n-1} F_{n-2}) for (n geq 2). This sequence generates a variety of interesting patterns and properties. One such intriguing property involves the partial sums of this sequence and their divisibility by 11. Specifically, finding the smallest value of (n) for which both the (n)-th and ((n-1))-th Fibonacci partial sums are divisible by 11 can lead us on a fascinating exploration of number theory.

Understanding Fibonacci Partial Sums

The partial sum (S_n) of the first (n) Fibonacci numbers is defined as:

[S_n F_0 F_1 F_2 ldots F_n]

The surprising fact is that the (n)-th partial sum can be succinctly expressed as:

[S_n F_{n 2} - 1]

Therefore, to satisfy the conditions that both (S_n) and (S_{n-1}) are divisible by 11, we must find (n) such that:

[F_{n 2} equiv 1 pmod{11}]

Exploring Fibonacci Numbers Modulo 11

To uncover the regularity in the Fibonacci sequence modulo 11, we compute the sequence modulo 11:

[ begin{align*} F_0 0 F_1 1 F_2 1 F_3 2 F_4 3 F_5 5 F_6 8 F_7 13 equiv 2 pmod{11} F_8 21 equiv 10 pmod{11} F_9 33 equiv 0 pmod{11} F_{10} 54 equiv 10 pmod{11} F_{11} 87 equiv 9 pmod{11} F_{12} 144 equiv 1 pmod{11} F_{13} 233 equiv 2 pmod{11} F_{14} 377 equiv 6 pmod{11} F_{15} 610 equiv 4 pmod{11} F_{16} 987 equiv 10 pmod{11} F_{17} 1597 equiv 7 pmod{11} F_{18} 2584 equiv 3 pmod{11} F_{19} 4181 equiv 0 pmod{11} F_{20} 6765 equiv 3 pmod{11} F_{21} 10946 equiv 8 pmod{11} F_{22} 17711 equiv 10 pmod{11} F_{23} 28657 equiv 6 pmod{11} F_{24} 46368 equiv 5 pmod{11} F_{25} 75025 equiv 0 pmod{11} F_{26} 121393 equiv 5 pmod{11} F_{27} 196418 equiv 6 pmod{11} F_{28} 317811 equiv 10 pmod{11} F_{29} 514229 equiv 3 pmod{11} F_{30} 832040 equiv 0 pmod{11} F_{31} 1346269 equiv 3 pmod{11} F_{32} 2178309 equiv 6 pmod{11} F_{33} 3524578 equiv 10 pmod{11} F_{34} 5702887 equiv 4 pmod{11} F_{35} 9227465 equiv 0 pmod{11} end{align*} ]

Now, let's check the conditions for the smallest (n):

If (n 10):?(F_{12} equiv 1), (F_{13} equiv 2) (not satisfied) If (n 11):?(F_{13} equiv 2), (F_{14} equiv 6) (not satisfied) If (n 12):?(F_{14} equiv 6), (F_{15} equiv 4) (not satisfied) If (n 13):?(F_{15} equiv 4), (F_{16} equiv 10) (not satisfied) If (n 14):?(F_{16} equiv 10), (F_{17} equiv 7) (not satisfied) If (n 15):?(F_{17} equiv 7), (F_{18} equiv 3) (not satisfied) If (n 16):?(F_{18} equiv 3), (F_{19} equiv 0) (not satisfied) If (n 17):?(F_{19} equiv 0), (F_{20} equiv 3) (not satisfied) If (n 18):?(F_{20} equiv 3), (F_{21} equiv 8) (not satisfied) If (n 19):?(F_{21} equiv 8), (F_{22} equiv 10) (not satisfied) If (n 20):?(F_{22} equiv 10), (F_{23} equiv 6) (not satisfied) If (n 21):?(F_{23} equiv 6), (F_{24} equiv 5) (not satisfied) If (n 22):?(F_{24} equiv 5), (F_{25} equiv 0) (satisfied)

Thus, the smallest (n) for which both (S_n) and (S_{n-1}) are divisible by 11 is 22.

Conclusion

The exploration of Fibonacci partial sums divisible by 11 demonstrates the intrinsic beauty of the Fibonacci sequence and its properties. The smallest (n) for which both (S_n) and (S_{n-1}) are divisible by 11 is 22. This discovery not only adds to our understanding of Fibonacci numbers but also highlights the elegance of number theory in uncovering hidden patterns and structures.