8191

8191 is the integer that equals 2^13 minus 1. It is a prime number and is classified as a Mersenne prime, corresponding to the exponent p = 13. Among known Mersenne primes, 8191 is the fifth in the sequence (3, 7, 31, 127, 8191, 131071, …).

In binary, 8191 is written as 1111111111111, a string of 13 consecutive ones. Its hexadecimal representation is 0x1FFF, and in octal it is 0o17777. As a 13-bit value, 8191 is also the maximum unsigned value representable with 13 bits, making it a common bitmask in certain computing contexts.

Mathematically, since 2^13 − 1 is prime, the associated even perfect number is 2^(13−1) × (2^13 − 1) = 2^12 × 8191 = 33,550,336. This demonstrates the link between Mersenne primes and perfect numbers: every Mersenne prime of the form 2^p − 1 yields an even perfect number when multiplied by 2^(p−1).

8191 therefore occupies a notable place in number theory as a small yet historically significant prime, illustrating the relationship between prime exponents, binary representations, and the generation of perfect numbers.