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Possible Assignment Topics or Starting PointsRemember:a) The main emphasis in this assignment is on your own explorations, rather than reading, so it isgood to choose a topic where you can do some exploring and conjecturing of your own before youread more about it. In some cases you may want to do quite a lot of reading at some point and findways to show that you have digested and made use of your reading whereas in others you may findyou do little reading.b) The nature of number theory is that you are unlikely to know where a number investigation maylead before you actually get involved in the maths, so be prepared to be taken in different directionsand end up with something rather different than you thought it would be. You may find that one ofthese starting points gives you enough (or more than enough) for your assignment. Or you may feelyou have got as far with one task as you can, and choose to start something different and submittwo pieces of work. Or you may want to try several to see which becomes most interesting, butinclude some work from the others. Or you may find that one task has links with another, and youcan 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 triviallysimple answer, or a challenging but accessible answer, or is a question that is still unanswered aftermany great mathematical minds have worked on it. So be prepared to adjust and adapt yourquestion 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 themhaving already done most of what you can do on your own, with plenty of questions.Topics1. Squares ContinuedContinue 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 drawingthem.You may want to link your results to Euclid's algorithm or continued fractions.2. How many in a rowExplore which rows of modular arithmetic multiplication tables have 2,3,4,5 etc different numbers inthem. What generalisations can you make?

3. Corners of SquaresDevelop the investigation we did in class on conjecturing and proving results about the corners of a2x2 square in the middle of an odd modular multiplication table to corners of a 3x3 squares in eventables, or 4x4 squares in odd tables, etc.4. Quadratic ResiduesInvestigate 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. DiagonalsInvestigate 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 TriplesInvestigate Pythagorean triples where two of the numbers are 2 apart. Extend to numbers 3, 4, 5apart etc. Can you make any generalisations?7. Unit denominator fraction investigationHow many ways can124be written as the sum of two unit fractions (fractions with one as theirnumerator)?Can you find a method for finding all the possibilities for any fraction1aand a way to predict howmany possibilities there will be for any a?Extension: How many ways can124be written as the difference of two unit fractions?