Item 41 (Gene Salamin): There are exactly 23,000 prime numbers less than 2^(18).
Item 154 (Bill Gosper): The myth that any given programming language is machine independent is easily exploded by computing the sum of powers of 2. If the result loops with period = 1 with sign +, you are on a sign-magnitude machine. If the result loops with period = 1 at -1, you are on a twos-complement machine. If the result loops with period greater than 1, including the beginning, you are on a ones-complement machine. If the result loops with period greater than 1, not including the beginning, your machine isn't binary -- the pattern should tell you the base. If you run out of memory, you are on a string or bignum system. If arithmetic overflow is a fatal error, some fascist pig with a read-only mind is trying to enforce machine independence. But the very ability to trap overflow is machine dependent. By this strategy, consider the universe, or, more precisely, algebra: Let X = the sum of many powers of 2 = ...111111 (base 2). Now add X to itself: X + X = ...111110. Thus, 2X = X - 1, so X = -1. Therefore algebra is run on a machine (the universe) that is two's-complement.
HAKMEM also contains some rather more complicated mathematical and technical items, but these examples show some of its fun flavor.
An HTML transcription of the document is available at ftp://ftp.netcom.com/pub/hb/hbaker/hakmem/hakmem.html.