# What's in a Number?

### Professor TIan-Xiao He explains the significance of 17.

Alex Bellos, author of popular math books, set up a survey website (at www.favouritenumber.net) seeking responses to the question, “What’s your favorite number and why?” We put the same question to Professor Tian-Xiao He, who responded that he finds 17 “among the most interesting” numbers. Here are just a few reasons why:

• 17 is the seventh prime number and sum of the first four primes. It is the first prime Leyland number, the third Fermat prime (as it is of the form 24+1), the third Stern prime, the third Pythagorean prime, the fourth Proth prime, the sixth Pierpont prime and the sixth Solinas prime. It is also the sixth Mersenne prime exponent yielding prime 131071 and the 13th term of the Euclid–Mullin sequence.
• 17 is the smallest Trotter prime (as it is of the form 10 n 2 +7 for n =1).
• 17 is the only known prime that is equal to the sum of the digits of its cube (because 173=4913 and 4+9+1+3=17; see sequence A046459).
• 17 is the only prime that is the average of two consecutive Fibonacci numbers.
• 17 is the only positive Genocchi number that is prime.
• 17 is the first number that can be written as the sum of a positive cube and a positive square in two different ways (that is, the smallest n such that x 3 + y 2 = n has two different solutions for positive integers x and y . The next such number is 65). It is also the smallest prime of the form p 3 + q 2 , where p and q are prime, and the only prime of the form pq + qp , where p and q are primes.
• If the edges are colored with three different colors, the number 17 is the minimum number of vertices on a complete graph such that there is bound to be a monochromatic triangle.
• At age 17 , German mathematician Carl Friedrich Gauss had a thought to construct the famous 17 -gon and finished it in the following year.
• Hungary-born mathematician Paul Erdös “gave a beautiful elementary proof of Bertrand’s conjecture (i.e., there is always a prime between any positive integer n and its double 2 n ) at the age of 17 , which is considered the first significant theorem he ever proved,” says He.
• At 17 years, the periodical cicada has the longest cycle of development of any known insect. Its Latin name given by Swedish naturalist Carl Linnaeus, Cicada septendecim, has 17 letters.
• The Parthenon has 17 columns along its lengths.
• Professor He’s home number is 1319 — the two closest prime numbers to 17 , i.e., 13 and 19.