Properties of the number 17
Seventeen is the seventh prime number, and a Fermat prime expressible as 2^2^2+1. Since it's a Fermat prime, a 17-sided polygon is constructible with compass and straightedge. For more on that see 257.
17 is also a Mersenne prime exponent, since 2^17-1 is prime.
17 is a record setter for period of digits in the reciprocal decimal expansion - its decimal expansion (0.0588235294117647, where the underlined digits repeat) repeats with a period of 16. See also 7 and 142,857.
17 is part of the formula x^2+x+17 which generates primes for -17<x<16 - a more wide ranged such formula is x^2+x+41 (see 41).
One of my favorite numerical properties of 17 is that it's the first number expressible as a sum of a square and cube in two ways (8+9 and 16+1). This is particularly appealing because one sum consists of 8 and 9, a unique pair of perfect powers that I discuss in the entry for 9, and the other consists of 16 and 1, two numbers with unique properties.
Also relating to powers, 17 = 9+8 = 92-82 = 34-43. This makes it the sum of one xy and yx pair (8 and 9) and the difference of another such pair (64 and 81). But note that the property 17 = 9+8 = 92-82, though it seems like a specil property, is not unique - it's always true that x+(x+1) = (x+1)2-x2.
The first 1000 digits of pi have exactly seventeen 17's.
The smallest duodecimal emirp (a prime that, when you reverse the digits, becomes another prime) is 17 in decimal, or 15 in duodecimal. Reversing 15 in duodecimal gives 51 in duodecimal, which is 61 in decimal, another prime. The smallest emirp pairs (for eample, 13 and 31 in decimal) in bases 2-16 are:
17 appears several times in Tupper's "self-referential" formula, which is really more of a magic trick than a self-referential formula - see Tupper's number, the 544-digit number that makes the formula work.
17 in Culture
17 is notable for sounding "psychologically random", seeing as it is used a lot less in many contexts than other numbers - such numbers usually end in 1, 3, 7, or 9, since even numbers and numbers that end in 5 are seen more often in daily life. Therefore 17 is an example of a number people would choose as a random number, and thus is occurs disproportionately often in film and literature.
Related to being psychologically random, 17 is also an example of a cult number, a number with a group of people who consider it special for a reason, and give it a certain special significance. Cult numbers have a tendency to be psychologically random, like 23, 27, and 37 for instance - some other cult numbers include pi, 7, 42, 47, 69, 137, and 666. This website and this one are a few cult number sites focused on the number 17, although in 17's case the origin of the cult is indeterminate - compare it to cult numbers with a clear origin like 42, 137, or 666.
Alongside all the love given to 17, Italy seems outright strange - in Italy 17 is known for bad luck similarly to 13 in most of the Western world (e.g. Italian buildings have no 17th floor). This bad luck partly stems from Latin because 17 in Roman numerals is XVII, which can be rearranged to form "vixi", which in Latin means "I have lived", implying the meaning "I am dead". Some 17 cultists have noted that bad luck and called it very strange.
Seventeen had some significance to me as a kid (mostly relating to my birth) - enough to be put on my "Very Important Numbers" list after two. I said that not many people would consider 17 a V.I.N (very important number), but that I do, because I was "supposed" to be born on March 17 but I was born 17 days late, and maybe other things. Other connections to 17 are that I live in the 17th state in America to become a state and my high school graduation year is 2017. I was very wrong saying that "not many people consider 17 important" though - very many people consider 17 a special number, as it's a notable cult number.
In fact, I have officially declared 17 my favorite two digit number, just as 7 is my favorite one digit number. It's also my own favorite cult number. As a kid I would've been surprised that seventeen has such a large cult following. If I discovered 17's popularity as a kid, I could have become obsessed with the number 17.
Such connections like I discussed are common among people - for example, Robert Munafo has similar connections with 27. I think numbers like 17 and 27 are more likely to have personal connections with people because such numbers are psychologically random (i.e. they don't occur as often as other numbers like 16 and 20), and therefore are noticed more often when they do occur - this once again connects psychologically random numbers with cult numbers.
Seven and Seventeen
Seven and seventeen (my favorite one-digit and two-digit numbers) make a pretty cool pair - they have similar names and are both prime, but even better: 7 is a Mersenne prime (a prime expressible as 2^n-1), while 17 is a Fermat prime (a prime expressible as 2^2^n+1). Better still, seventeen is the seventh prime number, and the prime numbers less than or equal to 7 (2+3+5+7) add up to 17.
Mersenne and Fermat primes are so similar, but so different - they have similar formulas, but are used in very different things. Fermat primes have many connections to mathematics (like constructibility of polygons and Pascal's triangle), but there are probably only five Fermat primes. Mersenne primes, on the other hand, have few connections to mathematics (other than a connection to perfect numbers), but there are many very large Mersenne primes that have millions of digits, making Mersenne primes very famous in googology. In fact, the largest known prime (with the exception of in 1951) has almost always been a Mersenne prime - the current one (as of 2014) has over 17 million digits.
Fun fact: This entry originally only said "Another prime and relatively uncommon number", and look what a monstrosity of an entry it's turned into today :)
(my favorite thing about 17 is that it's written 10 in base 17)
(edit 2018: did I also steal that joke from Robert Munafo? hell if I know...)