65,537

65,537 (sixty-five thousand five hundred and thirty-seven) is an integer. It holds significance in various fields, most notably in computer science and mathematics.

Mathematical Properties:

• It is a Fermat number, specifically F₄, where Fₙ = 2^(2ⁿ) + 1. Fermat numbers are integers of the form 2 raised to a power of 2, plus 1. The first few Fermat numbers are known to be prime, leading Pierre de Fermat to conjecture that all Fermat numbers are prime. However, this was later disproven. 65,537 is prime.
• It is the largest known Fermat prime. While many larger Fermat numbers are known, none have been proven to be prime.
• It is related to the constructible regular polygons. A regular n-gon is constructible with compass and straightedge if and only if n is the product of distinct Fermat primes and a power of 2.

Computer Science Significance:

• It is a common limit for certain data types, especially in older systems or embedded systems with limited memory. Specifically, it's often encountered as the maximum value that can be stored in a 16-bit unsigned integer (2¹⁶ - 1).
• Because of its relationship to 2¹⁶, it is frequently encountered in contexts involving 16-bit address spaces or memory limitations.
• In cryptography, its primality has made it relevant in some key generation algorithms, though it is less commonly used in modern cryptographic systems compared to larger primes.