65537

65537 is the integer after 65536 and before 65538.

[edit] In mathematics

65537 is a Fermat number, being 2¹⁶ + 1. It is also a Fermat prime, a Pierpont prime, and it is possible to construct with compass and straightedge a regular polygon with this many sides. Such a construction was devised by J. Hermes in the 19th century. See constructible polygon.

65537 is commonly used as a public exponent in the RSA cryptosytem. 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'.

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