For a regular 17-gon, the formulas above give the x-coordinate of the first vertex in the upper half plane. The first formula goes back to Gauss. The second formula is obtained by a more elementary method, see pages 39 to 47 in this book. This method uses only the trigonometric addition theorem and some clever guesses. It needed some optimism to create this book about number theory. The proofs are gapless and readable, and there are given some exercises with solutions and algorithms. Especially the geometric construction of the regular 17, 257 and even the 65 537-gon are treated in complete and purely constructive details, including programming codes. Otherwise could be covered just an important classical selection.