You’re visiting Carnival of Mathematics #172 hosted by Cassandra Lee. Find all Carnivals here and the previous Carnival (#171) here. Many thanks to the Aperiodical for the wonderful opportunity to host this Carnival.
The Roman representation of the number 172 comprises the initials of my English name, the chromosomes determining my gender and the eating utensils that I use daily. Yes, that’s CLXXII. *gasps at the sudden weight on her shoulders*
Without further ado, let’s begin with something light-hearted. It’s easy to show that 172 = 2 × 2 × 43, an even composite number; it doesn’t appear special until you also learn that 172 is also a deficient number (the sum of the factors is less than the number itself: 2 + 2 + 43 = 47 < 172) and an evil number (!), which means it has an even number of 1’s in its binary representation (17210 = 101011002). Note that, if the units digit is relabelled the “zeroth/0-th” digit, then you can verify that the binary representation of 172 (= 101011002) has the following property:
The n-th digit is 1 if and only if n is prime.
Furthermore, 172 is in the lazy caterer’s sequence at n = 18; the maximum number of pieces you can get with 18 cuts of a circular pie is 172.