For any prime number p except 2 and 5, either Fp-1 or fp+1 is divisible by p. If I showed that this is true for p=7, the answer should be F8---is this correct because f8 is 21.
P=11 would be divisible by 55 which is F10.
Looks good.
To verify if the statement is true, we can use the properties of the Fibonacci sequence and modular arithmetic.
Let's refer to the statement as: For any prime number p, except 2 and 5, either Fp-1 or Fp+1 is divisible by p.
To prove the statement, we need to show that for any prime number p (except 2 and 5), either Fp-1 or Fp+1 is divisible by p.
Let's start by considering p = 7:
Step 1: Calculate the Fibonacci number at index p = 8: F8 = F7 + F6 = 13 + 8 = 21.
Step 2: Check if either Fp-1 or Fp+1 is divisible by p:
- F7 = 13 is not divisible by 7.
- F9 is the Fibonacci number after F8, so let's calculate it using the Fibonacci sequence: F9 = F8 + F7 = 21 + 13 = 34. F9 is not divisible by 7.
Based on this calculation, for p = 7, neither Fp-1 nor Fp+1 is divisible by p. Hence, we cannot conclude that the result is F8 = 21 for p = 7.
Let's consider p = 11:
Step 1: Calculate the Fibonacci number at index p = 10: F10 = F9 + F8 = 34 + 21 = 55.
Step 2: Check if either Fp-1 or Fp+1 is divisible by p:
- F9 = 34 is not divisible by 11.
- F11 is the Fibonacci number after F10, so let's calculate it using the Fibonacci sequence: F11 = F10 + F9 = 55 + 34 = 89. F11 is not divisible by 11.
Based on this calculation, for p = 11, neither Fp-1 nor Fp+1 is divisible by p. Hence, we cannot conclude that the result is F10 = 55 for p = 11.
Therefore, the statement isn't true for p = 7 or p = 11. It's important to note that this statement may not hold true for all prime numbers p (except 2 and 5) and might require further investigation to determine its validity.