University Number Theory Assignment: Explorations and Proofs

This assignment provides a comprehensive exploration of number theory, offering a diverse range of topics for investigation. Students are encouraged to delve into areas such as squares, modular arithmetic, quadratic residues, Pythagorean triples, continued fractions, consecutive sums, and unit fractions, among others. The assignment emphasizes independent exploration, conjecturing, and the development of proofs. Students can select from a list of starting points, adapting and adjusting their questions as they progress. The goal is to encourage a deep understanding of number theory concepts and the ability to formulate and prove mathematical results. Desklib is a platform that provides past papers and solved assignments for students to aid their learning and understanding of complex topics.

Possible Assignment Topics or Starting Points
Remember:
a) The main emphasis in this assignment is on your own explorations, rather than reading, so it is
good to choose a topic where you can do some exploring and conjecturing of your own before you
read more about it. In some cases you may want to do quite a lot of reading at some point and find
ways to show that you have digested and made use of your reading whereas in others you may find
you do little reading.
b) The nature of number theory is that you are unlikely to know where a number investigation may
lead before you actually get involved in the maths, so be prepared to be taken in different directions
and end up with something rather different than you thought it would be. You may find that one of
these starting points gives you enough (or more than enough) for your assignment. Or you may feel
you have got as far with one task as you can, and choose to start something different and submit
two pieces of work. Or you may want to try several to see which becomes most interesting, but
include some work from the others. Or you may find that one task has links with another, and you
can find a way to write an assignment that links the two.
c) The nature of number theory is that it is hard to know whether a simple question has a trivially
simple answer, or a challenging but accessible answer, or is a question that is still unanswered after
many great mathematical minds have worked on it. So be prepared to adjust and adapt your
question as you explore in order to find something you can work on at your level.
d) Tutorials are intended to help you to go further than you can on your own. So come to them
having already done most of what you can do on your own, with plenty of questions.
Topics
1. Squares Continued
Continue with the squares investigation from the first lesson, exploring patterns in a table of either:
 how many different sizes of square;
 how many squares altogether.
Use your patterns to predict how many squares there will be in bigger rectangles without drawing
them.
You may want to link your results to Euclid's algorithm or continued fractions.
2. How many in a row
Explore which rows of modular arithmetic multiplication tables have 2,3,4,5 etc different numbers in
them. What generalisations can you make?

3. Corners of Squares
Develop the investigation we did in class on conjecturing and proving results about the corners of a
2x2 square in the middle of an odd modular multiplication table to corners of a 3x3 squares in even
tables, or 4x4 squares in odd tables, etc.
4. Quadratic Residues
Investigate quadratic residues for non-prime n.
Can you say anything about:
a) how many different quadratic residues there are;
b) how many different numbers give the same value when squared;
c) whether 0 is the square of a non-zero value;
d) whether -1 is a quadratic residue.
5. Diagonals
Investigate the relationship between the two diagonals of the modular multiplication table.
When do the following occur:
a) Both contain the same numbers;
b) All numbers on one diagonal are different from all numbers on the other.
c) Some numbers are the same and some are different.
6. Two-apart Pythagorean Triples
Investigate Pythagorean triples where two of the numbers are 2 apart. Extend to numbers 3, 4, 5
apart etc. Can you make any generalisations?
7. Unit denominator fraction investigation
How many ways can 1
24 be written as the sum of two unit fractions (fractions with one as their
numerator)?
Can you find a method for finding all the possibilities for any fraction 1
a and a way to predict how
many possibilities there will be for any a?
Extension: How many ways can 1
24 be written as the difference of two unit fractions?

Can you find a method for finding all the possibilities for any unit fraction 1
a ?
8. Quadratic expressions
Investigate the expression a2+2b2. Which numbers can be written in this way for two different pairs
of values of a and b? Can some be written in three or four ways? Is there a way to use complex
numbers to help with this? Which numbers cannot be written like this at all?
Repeat for other quadratic expressions in a and b, eg a2+3b2 , a2+ab+b2.
The next three tasks (9,10,11) start with well-known 'investigations' often used in classrooms and,
in the past, as GCSE coursework. It is worth taking time to play with the initial stages to get a feel
for the problem (you will probably want to use it in your own teaching). I would also suggest you
resist at first looking it up on the internet (though there will be a lot there about the stages
relevant to secondary school work). What will eventually make this level 6 university work is the
extension questions and how you develop them.
9. Consecutive Sums
21 can be made by the addition of consecutive numbers in the following ways
1+2+3+4+5+6 6+7+8 10+11
Explore other numbers and their consecutive sums. Which numbers do not have a consecutive sum?
Which numbers have more than one consecutive sum?
How many consecutive sums are there for 49?
Investigate for numbers other than 49. Explore numbers that can be consecutive sums in two, three,
four etc ways.
10. Stamps:
You have a large sheet of 5p stamps, and a large sheet of 7p stamps.
What values can you make with them (eg you can make 17p with two 5ps and one 7p).
What values can't you make? Generalise.
Extension: I have two sheets of two different values of stamps. 43p is the largest number that I
cannot make. What could the values of the stamps be?
Investigate for numbers other than 43. Which can be the largest unmakeable number for more than
one pair of stamp values?

11. Frogs
From: http://www.hellam.net/maths2000/frogs.html which has an interactive version to play with.
Extension: A puzzle takes 43 moves altogether to swap the frogs. How many frogs of each colour could there have been?Investigate for
numbers other than 43. Which have two or more pairs of possible different numbers of frogs of each colour that would give that number
of moves? Can you generalise?
12. Square Roots and Continued Fractions
Explore the length of the repeating part of a continued fraction for a square root. Can you make any
predictions or generalisations?
13.Socks
Investigate the following problem. How does it link to things we have studied?
My daughter keeps her odd socks in many strange places.
For one drawer, the probability that the first two socks she picks at random are
both purple is exactly ½ .
How many socks are there in the drawer? How many are
purple?
Is there more than one answer?
Investigate variations of the problem.
14. Variations on Pell
Investigate integer solutions to a2-10b2 = -1.
For different values of d and e explore the equation a2-db2 = e.

Identify some cases that have no solutions. Can you prove why?
Use computer searches and/or other methods to find solutions for some other cases.