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Theory of dividing the circle into equal parts or constructing regular polygons or, analytically, of finding the n th roots of unity.



the division of a circle into n equal parts, which is one of the oldest problems of mathematics. This problem consists in dividing a circle using only a compass and straightedge. Ancient Greek mathematicians were able to divide a circumference into 3,5, and 15 parts and to double indefinitely the number of sides of the resulting polygons. At the end of the 18th century, K. Gauss demonstrated that a circle could be divided, using a compass and straightedge, into 17 parts and into n parts where n is a prime of the form n = 22k + 1 or a product of such primes and an arbitrary power of 2 (for k = 0, 1, 2, 3,4, we get the prime numbers n = 3, 5, 17, 257, 65, 537; for k = 5, 6, 7, the corresponding numbers are not prime). It is impossible, using a compass and straightedge, to divide a circle into any other number of equal parts. The problem of dividing a circle is equivalent to solving the binomial equation xn - 1 = 0. It is possible to divide a circle using a compass and straightedge only when the roots of this equation can be obtained by the successive solution of quadratic and linear equations.

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