8501

This number is a prime.

The prime p=11 has an unusual property: the number 2^p - 1 (= 2047) and the pth Fibonacci number (89) have a common divisor (in this case, 89). The next largest prime with this property is 8501. 2^8501 - 1 is a number with 2560 digits, while the 8501th Fibonacci number has 1777 digits, and those two large numbers are both divisible by 680081. [Keith]