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FAMOUS PROBLEMS OF MATHEMATICS
There is still one unresolved question-for which values of m is 2 z• + 1 a prime number? We know that Fermat's number is prime for m = 0, 1,2,3,4, and in fact, we have constructed regular polygons of 3, 5, and 17 sides. For m = 3, n = 257; and for m = 4, n = 65,537, the analysis has also been accomplished. L. E. Dickson, in a discussion of "Constructions With Ruler and Compasses" which appears in Monographs on Topics of Modern Mathematics, states "The regular 257-gon has been discussed at great length by Richelot in Crelle's Journal fur Mathematik, 1832; and geometrically by Affolter and Pascal in Rindicinti della R. Accademia di Napoli, 1887. The regular polygon of 2 16 + I = 65,537 sides has been discussed by Hermes; Gottingen Nachridten, 1894." Also, in the April 1961 issue of Scientific American, Martin Gardner describes some of the topics discussed by H. M. S. Coxeter in a recently published book An Introduction to Geometry. Martin Gardner quotes Professor Coxeter to the effect that there is at the University of Gottingen a large box containing a manuscript showing how ro construct a regular polygon of 65,537 sides. Gardner also writes "a polygon with a prime number of sides can be constructed in the classical manner only if the number is a special type of prime called a Fermat prime; a prime that can be expressed as 2 z' + I. Only five such primes are known-3, 5, 17, 257, 65,537. The poor fellow who succeeded in constructing the 65,537-gon, Coxeter tells us, spent ten years in the task." As Euler showed, when m = 5, 2 z• + 1 has a factor 641. Dickson states that for n = 6, 7, 8, 9, the number is not prime. For the next case n = 10, it has not been determined whether or not this Fermat number is prime. It may very well be that 2 z• + 1 is prime only for values of m < 5. In that case the formula indicating which polygons are constructible would read