I have encountered the number 17 several times in the last few weeks—enough times that it caught my attention. So I challenged myself to write a list of seventeen interesting things about the number 17. I tried to be as mathematical as possible. I wasn’t able to get seventeen facts on my own, so I turned to the internet. As it turns out, there are quite a few web pages about the number 17 (shocker, I know!). In particular, it turns out that a mathematics professor at Hampshire College, David Kelly, has been lecturing about 17 for a while. When he retired, the college changed all of the 15 MPH speed limit signs to 17 MPH.

1. $17=2^{2^{2}}+1$ is a Fermat prime.
2. The teenage Gauss proved that the regular 17-gon is constructible by compass and straightedge (which is related to the previous bullet).
3. There are 17 wallpaper groups.
4. A haiku has 17 syllabes.
5. A Sudoku needs at least 17 clues to have a unique solution.
6. Theodorus proved that each of $\sqrt{3},$ $\sqrt{4},$ $\sqrt{5},$…, $\sqrt{17}$ is an integer or is irrational. (Actually, the wording in Plato’s Theaetetus is ambiguous; it could have been that $\sqrt{17}$ is the first one that Theodorus was unable to show was irrational.)
7. According to hacker lore, 17 is the “least (or most) random number.”
8. To the nearest order of magnitude, the universe is $10^{17}$ seconds old (approximately $4.32\times 10^{17}$ seconds).
9. It is the smallest number that is the sum of two distinct positive integers raised to the fourth power: $17=1^4+2^4.$
10. It is the smallest number that can be written as the sum of a square and a cube in two different ways $17=3^2+2^3=4^2+1^3.$
11. Some cicadas have a 17-year life cycle.
12. There are 17 ways to write 17 as the sum of primes.
13. The Italians think 17 is unlucky (apparently because XVII can be rearranged to be VIXI, which means “my life is over”).
14. Plutarch wrote “The Pythagoreans also have a horror for the number 17, for 17 lies exactly halfway between 16, which is a square, and the number 18, which is the double of a square, these two, 16 and 18, being the only two numbers representing areas for which the perimeter equals the area.”
15. There are 17 nonabelian groups of order at most 17.
16. 17 is the smallest whole number whose reciprocal contains all ten digits: $\frac{1}{17}=0.\overline{0588235294117647}.$
17. In Ramsey theory $R(3,3,3)=17.$ In other words, when $n\le 16$ it is possible to color the edges of any graph with vertices using three colors so that there are no monochromatic triangles. But this is impossible for $n=17$ (the complete graph with 17 vertices can’t be colored in this way). I don’t think this was what Stevie Nicks was signing about in her song “Edge[s] of [K_]Seventeen.”

1. Japheth Wood says:

Reports of David Kelly’s demise are premature! He’s still alive and well, planning out this summer’s HCSSiM, and a special reunion, YPMD ’17.

1. Yikes! I read the article too quickly apparently. Thanks. Fixed.

2. It is interesting to note that √ [17] = 4.123105626

Subtracting the number 4 from it we get:

0.123105626

Inverse of this number is:

8.123105625

Subtracting 4 we get back to:

4.123105626 = √ [17]

Panagiotis Stefanides

http://www.stefanides.gr