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(Hint: consider expressions of the form 12x2+ 1.) Question 3 Let a ∈ N

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FINAL EXAM ‘INLEIDING IN DE GETALTHEORIE’

Thursday, 5th January 2016, 13.30 pm - 16.30 pm

Question 1

a) Find the continued fraction expansion of √ 33.

b) What number has the continued fraction expansion [5, 1, 4, 1, 10, 1, 4, 1, 10, ...] ? Question 2

Show that there is an infinite number of primes of the form p = 6m + 1 with m ∈ N. (Hint: consider expressions of the form 12x2+ 1.)

Question 3

Let a ∈ N. Assume that 

a p



= 1 for every odd prime number p - a. Show that then a has to be a square number.

Question 4

Give a proof of the identity X

d|n

(−1)n/dφ(d) =

 0 if n ∈ N is even

−n if n ∈ N is odd.

Question 5

Is there a natural number n ∈ N such that d(n) = 7 where d(n) is the divisor function? Is there a natural numer n such that φ(n) = 7?

Question 6

A Pythagorean triangle is a triangle with one right angle and such that all the sides have integer length. We say that a pythagorean triangle has consec- utive legs, if the difference between the two shortest sides is exactly equal to one. Find at least 2 different pythagorean triangles with consecutive legs and show how pythagorean triangles with consecutive legs are related to solutions of a certain Pell’s equation.

Note: A simple non-programmable calculator is allowed for the exam.

Date: 5th January 2016.

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