65537 is the integer after 65536 and before 65538.

In mathematics

65537 is the largest known prime number of the form , where . In number
theory, primes of this form are known as Fermat primes, named after the mathematician
Pierre de Fermat

Pierre de Fermat

Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

. To date, the only prime Fermat numbers are









and the largest known Fermat prime,



In 1732, Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 found that the next Fermat number is composite:



In 1880, F. Landry showed that

Applications

65537 is commonly used as a public exponent in the RSA cryptosystem. This value is seen as a wise compromise, since it is famously known to be prime, large enough to avoid the attacks to which small exponents make RSA vulnerable, and can be computed extremely quickly on binary computers, which often support shift and increment instructions. Exponents in any base can be represented as shifts to the left in a base positional notation system, and so in binary the result is doubling - 65537 is the result of incrementing shifting 1 left by 16 places, and 16 is itself obtainable without loading a value into the register (which can be expensive when register contents approaches 64 bit), but zero and one can be derived more 'cheaply'.