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Seventeen is the 7th prime number. The next prime is nineteen, with which it forms a twin prime. 17 is the sum of the first four primes. 17 is the sixth Mersenne prime exponent, yielding 131071. 17 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 17 is the third Fermat prime, as it is of the form 24 + 1, and it is also a Proth prime. Since 17 is a Fermat prime, heptadecagons can be drawn with compass and ruler. This was proven by Carl Friedrich Gauss.[1] Another consequence of 17 being a Fermat prime is that it is not a Higgs prime for squares or cubes; in fact, it is the smallest prime not to be a Higgs prime for squares, and the smallest not to be a Higgs prime for cubes. 17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime. As 17 is the least prime factor of the first twelve terms of the Euclid–Mullin sequence, it is the thirteenth term. Seventeen is the aliquot sum of two numbers, the odd discrete semiprimes 39 and 55 is the base of the 17-aliquot tree. There are exactly seventeen two-dimensional space (plane symmetry) groups. These are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper. Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, for all positive n < p − 1. In the Irregularity of distributions problem, consider a sequence of real numbers between 0 and 1 such that the first two lie in different halves of this interval, the first three in different thirds, and so forth. The maximum possible length of such a sequence is 17 (Berlekamp & Graham, 1970, example 63). Either 16 or 18 unit squares can be formed into rectangles with perimeter equal to the area; and there are no other natural numbers with this property. The Platonists regarded this as a sign of their peculiar propriety; and Plutarch notes it when writing that the Pythagoreans "utterly abominate" 17, which "bars them off from each other and disjoins them".[2] 17 is the tenth Perrin number, preceded in the sequence by 7, 10, 12. In base 9, the smallest prime with a composite sum of digits is 17. 17 is known as the Feller number, after the famous mathematician William Feller who taught at Princeton University for many years. Feller would say, when discussing an unsolved mathematical problem, that if it could be proved for the case n = 17 then it could be proved for all positive integers n. He would also say in lectures, "Let's try this for an arbitrary value of n, say n = 17." Similar to Feller, Prof. Vadim Khayms of Stanford University is also known to use 17 as an arbitrary value during lectures. His Computational Mathematics for Engineers course includes 17 lectures. 17 is the least random number,[3] according to the Hackers' Jargon File. There is a proven theorem that 17 is the value most likely to be picked as a "random" number when such is needed in journalism which is derived from the Feller number.[4] It is a repunit prime in hexadecimal (11). It is believed that the minimum possible number of givens for a sudoku puzzle with a unique solution is 17, but this has yet to be proven. There are 17 orthogonal curvilinear coordinate systems (to within a conformal symmetry) in which the 3-variable Laplace equation can be solved using the separation of variables technique. 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 x3 + y2 = n has two different solutions for x and y positive integers. The next such number is 65.