August 27, 2012 | 3
On Tuesday, Scientific American turns 167 years old. It doesn’t exactly look like the kind of anniversary we usually celebrate, with our decimal normative number system that overvalues ending zeroes and fives, but 167 is a pretty neat number. First of all, we can insert two symbols into it to get a correct mathematical statement: 1 + 6 = 7. Isn’t that nice? But that’s not all.
Perhaps most appropriately for our birthday party, 167 is a happy number. (Yes, it’s a technical term.) The origin of the term is unknown, but to determine whether an integer is happy, we just keep repeating the process of adding up the squares of the digits. If we eventually reach the number 1, the number is happy. So for 167, we get 12 + 6 2 + 72 = 1 + 36 + 49 = 86. Then we repeat: 82 + 62 = 64 + 36 = 100. We’re almost there: 12 + 02 + 02 = 1. We did it! We won’t be happy again until 2021, our 176th anniversary (note that the numbers are numerical anagrams of the current year and anniversary, respectively), and only 15.5 percent of numbers between 1 and 10122 are happy, so we should enjoy it while we can.
If we expand our horizons past the constraints of our base 10 number system, we can play with 167 even more. Any time you write down a number, you are implicitly using a base, whether you realize it or not. Usually we use base 10, which means that the furthest right place in the number is the ones place (100), the next place to the left is the tens place (101), the next is the hundreds place (102), and so on. But any number can replace 10 in the above description. For example, computers use base 2, or the binary system. To write 167 in base two, we first write it as a sum of powers of two: 167 = 128 + 32 + 4 + 2 + 1. This becomes 27 + 25 + 22 + 21 + 20, which is 10100111 in binary.
So maybe we should explore different bases and see if 167 looks better in any of them. Occasionally someone will advocate the duodecimal (base 12) system, and it already shows up a bit in our clocks, rulers and egg cartons. But 167 = 11B in base 12, which sounds more like an apartment in a wacky sitcom than a respectable number. (For bases larger than 10, we use letters to represent the numbers 10, 11 and so on, since we don’t have any numerals left. B refers to 11.) Base 16, or hexadecimal, which along with binary is used in computer programming, is not much better: A7.
Maybe we can find something better by searching bases smaller than 10. Then we definitely won’t be stuck with any letters in our numbers. The number 167 is five more than a multiple of nine, so in some bases that are divisible by nine, it ends with a five, which looks much more celebratory than that gawky 7. (Of course, it only looks special to our decimal-accustomed eyes; the number 5 would not have the significance it does if we thought in base 9.) In base 9, 167 is written 205, but I personally prefer 25 in base 81. It’s simple and svelte. (It’s also how old Scientific American tells people she is. It’s not a lie, it’s an omission of base.)
Looking at 167 in terms of different bases yields another fun fact: 167 is a strictly nonpalindromic number, meaning that it can’t be written as a palindrome (the same forwards and backwards) in any base between 2 and 165. (The reason we stop at base 165, which is 167-2, is that every number n is a palindrome in base n-1. It will always look like 11.) So far the percentage of integers that are strictly nonpalindromic is unknown, but the next one after 167 is 179, and after that we’ll have to wait until we’re 223.
In addition to the traits listed above, which are surely more than enough to warrant a celebration, 167 is, among other things, a safe prime, a highly cototient number and a full reptend prime. I really like the last one: it implies that there is a 166-digit-long number such that every multiple of it is a cyclic permutation of the digits. That is, when you multiply it by any other integer, the digits are exactly the same and in the same order but started in a different place, like 142857 x 2 = 285714. (142857 is the smallest nonsilly example.)
Scientific American is thrilled to be celebrating her 25th (*coughbase81cough*) birthday! What’s your favorite thing about the number 167?