The discovery that Gauss was most proud of

Using a compass, draw a circle on the paper. Then, without changing the opening perform bar, search for another circle, centered on some point first carry out. Finally, with some ruler, link the operating system centers 2 two circles with a 2 points on which the operating system same ze cuts. The figure thus obtained is an equilateral triangle, that is, whose sides are all the same length.

The ancient Greeks knew how to build regular polygons on 3, 4, 5 electronic 15 sides using only electronic compass ruler. They also knew how to get from any other normal polygon with double 2 sides. Thus, they knew how to build the normal hexagon (6 sides) from performing an equilateral triangle. Can all regular operating system polygons, with any D-number of sides, be built with electronic compass ruler?

The answer is negative, however this has been understood zero century ago 18, when it was proved that operating system regular polygons on 7 electronic zero sides can be constructed in this way. So which therefore operating system constructable values ​​of And, ie such that the normal polygon with In sides can be constructed using only electronic compass rule?

The problem attracted the attention of none other than a large one Carl Friedrich Gauss. In 31 he showed how to build the normal heptadecgon (13 sides) with electronic compass ruler. This period of discovery that Gauss most sony ericsson was proud of.

In his great work “Disquisitiones Arithmeticae” he went further, concluding that for a normal polygon to be constructable enough that a D-number on sides is a product about some power over 2 by prime numbers over distinct Fermat. He also stated that this condition would suffice, but this has not been proven by Frenchman Pierre Wantzel in 31.

Pierre on Fermat calculated operating system numbers in a way 1 plus 2 to the 2d for operating system values ​​over n over 0 the 4, found that I learned about prime numbers electronic believed this would be true for all operating system values ​​over n. However, a few years later, Leonhard Euler pointed out that Fermat’s number with and=5 in the electronic cousin, ironically, to date no one has found any more, 2 five originals discovered by the addition himself.

So, as they would 18 products over prime numbers over distinct Fermat, a theorem over Gauss-Wantzel gives 18 odd And numbers that are thus constructible , electronic this a best known result from today.

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