By Votja P. A.

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**Extra resources for Diophantine Approximations and Value Distribution Theory**

**Sample text**

An important property of the series is that the ratios of successive pairs are closer and closer approximations to the golden number τ just mentioned in connection with Fermat numbers. 625; and so on. 618033989.... It has many neat properties such as τ – 1 = 1/τ and τ + 1 = τ2. We will encounter the Fibonacci numbers again in Lecture 4 on Fractions when we discuss Egyptian fractions and continued fractions, and in Lecture 8 on Primes when we discuss the greatest common divisor. They also turn up in Appendix B and Appendix G.

The basic reason why the constructions can be done is that we have to solve an equation of degree F–1. In the case of the 17-gon we thus have to solve an equation of degree 16. It turns out that this can be done by solving a series of nested quadratic equations since 16 = 24. The roots of a quadratic equation just involve square roots and these can be found by ruler and compasses alone. In the case of a pentagon we need to find the square root of 5 and this is easily done since the hypotenuse of a right-angled triangle whose other sides are 1 and 2 is of course √5 using Pythagoras as shown above.

Then find the factors of 14288 (its largest prime factor is 47) and show that they add to 15472. And similarly show that the factors of 15472 (largest prime is 967) add to 14536 and that those of 14536 (largest prime 79) add to 14264. Finally, show that the factors of 14264 (largest prime 1783) add to 12496 thus completing the cycle of 5. Show that the remainder on dividing an even perfect number (other than 6) by 3 is always 1. 3 Probability we look at some of the basic ideas of probability theory.