The discovery that Gauss was most proud of – 07/27/2021 – Marcelo Viana

Draw a circle on the paper with a compass. Then, without changing the opening of the compass, draw another circle centered somewhere in the first one. Finally, use a ruler to connect the centers of the two circles to one of the points where they intersect. The figure obtained in this way is an equilateral triangle, that is, the sides of which are all of the same length.

The ancient Greeks knew how to use a ruler and compass to build regular polygons with 3, 4, 5 and 15 sides. They also knew how to get from one regular polygon to another with double sides. So they knew how to build the regular hexagon (6 sides) from the equilateral triangle. Can all regular polygons with any number of N sides be constructed with a ruler and compass?

The answer is no, but that wasn’t understood until the 18th century when it was proven that regular 7- and 13-sided polygons couldn’t be constructed this way. So what are the constructible values ​​of N, ie so that the regular polygon with N sides can only be constructed with a ruler and compass?

The problem caught the attention of none other than the great Carl Friedrich Gauss. In 1796 he showed how to build the regular heptagon (17 pages) with a ruler and compass. This was a discovery Gauss was most proud of. In his great work “Disquisitiones Arithmeticae” he went even further and came to the conclusion that for the construction of a regular polygon it is sufficient that the number N of sides is the product of a power of 2 divided by different Fermat prime numbers. He also stated that this condition would be sufficient, but this was only proven in 1837 by the Frenchman Pierre Wantzel.

Pierre de Fermat calculated the numbers of the form 1 plus 2 to 2n for n values ​​from 0 to 4, found them to be prime numbers and held this for all n values. But a few years later Leonhard Euler pointed out that the Fermat number with n = 5 is not a prime number and ironically, until today it has found none other than the five originals he discovered.

Since there are 31 different Fermat prime number products, the Gauss-Wantzel theorem gives 31 odd N numbers that can be constructed, and this is the best result known so far.

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