65,536

Microsoft Excel held a maximum of 65,536 rows until the 2007 release raised it to 1,048,576 — that is, from 2¹⁶ to 2²⁰. The older ceiling had consequences beyond inconvenience: in 2020 Public Health England lost thousands of COVID-19 test results because case data was being collated in the legacy XLS format, whose row limit silently truncated the files. The limit was not a bug, and neither was the truncation; the failure was using a format whose documented ceiling was below the data volume.

A 16-bit field holds 65,536 distinct values, so the largest is 65,535 — which is why TCP and UDP port numbers stop there, why an unsigned 16-bit counter wraps at that point, and why so many older file formats cannot describe anything larger. The distinction between 65,536 and 65,535 is the same off-by-one that separates 256 from 255 and is the single most common source of confusion about both.

There are 65,536 values a 16-bit field can hold, which is also the size of the Unicode Basic Multilingual Plane.

Unicode originally allotted 16 bits per character, assuming 65,536 slots would cover every script. It did not, and rather than widen the default the standard added surrogate pairs — two 16-bit units encoding one character. That compromise is why text handling still breaks at this boundary, in emoji and in less common scripts alike.

65,537 is 2¹⁶ + 1 and is prime — the fifth and largest known Fermat prime, after 3, 5, 17 and 257. No further one has been found despite extensive search, and many mathematicians expect there are none. It is the standard public exponent in RSA because it is prime, large enough to be safe, and has only two set bits, which makes exponentiation fast. So the number nearly every encrypted connection uses is chosen for an arithmetic accident of the number one above a power of two.

65,536 = 2^2^2^2 — two raised to two raised to two raised to two — which makes it the fourth term of a sequence that grows faster than any ordinary notation can follow. The fifth term is 2^65536, a number with nearly twenty thousand digits, and the sixth cannot be written down in the observable universe. 65,536 is the last term of the sequence that fits comfortably in a sentence, which is a decent illustration of what tetration does.