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Concept
17 (number)
Summary
17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.
Seventeen is the sum of the first four prime numbers.
Seventeen is the seventh prime number, which makes it the fourth super-prime, as seven is itself prime. It forms a twin prime with 19, a cousin prime with 13, and a sexy prime with both 11 and 23. Seventeen is the only prime number which is the sum of four consecutive primes (2, 3, 5, and 7), as any other four consecutive primes that are added always generate an even number divisible by two. It is one of six lucky numbers of Euler which produce primes of the form , and the sixth Mersenne prime exponent, which yields 131,071. It is also the minimum possible number of givens for a sudoku puzzle with a unique solution. 17 can be written in the form and ; and as such, it is a Leyland prime and Leyland prime of the second kind:
17 is the third Fermat prime, as it is of the form with . On the other hand, the seventeenth Jacobsthal–Lucas number — that is part of a sequence which includes four Fermat primes (except for 3) — is the fifth and largest known Fermat prime: 65,537. It is one more than the smallest number with exactly seventeen divisors, 65,536 = 216. Since seventeen is a Fermat prime, regular heptadecagons can be constructed with a compass and unmarked ruler. This was proven by Carl Friedrich Gauss and ultimately led him to choose mathematics over philology for his studies.
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".
17 is the minimum number of vertices on a graph such that, if the edges are colored with three different colors, there is bound to be a monochromatic triangle; see Ramsey's theorem.
There are also:
17 crystallographic space groups in two dimensions.
Official source
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