• No results found

Question 2 Find the quadratic numbers that belong to the continued fractions

N/A
N/A
Protected

Academic year: 2021

Share "Question 2 Find the quadratic numbers that belong to the continued fractions "

Copied!
1
0
0

Bezig met laden.... (Bekijk nu de volledige tekst)

Hele tekst

(1)

3RD EXAM ‘INLEIDING IN DE GETALTHEORIE’

Tuesday, 25th October 2016, 9 am - 10 am

Question 1

Find the continued fraction of √

12 and √ 17.

Question 2

Find the quadratic numbers that belong to the continued fractions [3, 1, 6, 1, 6, 1, 6, 1, 6, ...] and [2, 1, 8, 1, 8, 1, 8, 1, 8, ....]

Question 3

Show that for any natural numbers p, q ∈ N one has

√5 −p q

> 1 5q2.

Question 4

Find at least two different solutions to the equation

1 + 2 + . . . + k = (k + 1) + (k + 2) + . . . + (l − 1) + l,

with k, l ∈ N and l > k and show how it is related to a Pell’s equation.

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

Date: 25th October 2016.

1

Referenties

GERELATEERDE DOCUMENTEN

For aided recall we found the same results, except that for this form of recall audio-only brand exposure was not found to be a significantly stronger determinant than

The convergent of an ordinary continued fraction can be computed by solving a tri- diagonal linear system for its first unknown. In this paper this approach is generalized to

is that every equivalence class contains exactly one reduoed form. In the real quadratic case, this is not true any more; here every equivalence class contains a whole oyole of

In conclusion, this thesis presented an interdisciplinary insight on the representation of women in politics through media. As already stated in the Introduction, this work

This could be done in fulfilment of the mandate placed on it by constitutional provisions such as section 25 of the Constitution of Republic of South Africa,

Gezien deze werken gepaard gaan met bodemverstorende activiteiten, werd door het Agentschap Onroerend Erfgoed een archeologische prospectie met ingreep in de

In werklikheid was die kanoniseringsproses veel meer kompleks, ’n lang proses waarin sekere boeke deur Christelike groepe byvoorbeeld in die erediens gelees is, wat daartoe gelei

ABSTRACT We prove that the numerators and denommators of the convergents to a real irrational number θ satisfy a linear recurrence with constant coeffi- cients if and only if θ is