# Possible Number Theory Investigations

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3. Corners of SquaresDevelop 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 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, 5 apart etc. Can you make any generalisations?7. Unit denominator fraction investigationHow many ways can 124 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 1a and a way to predict how many possibilities there will be for any a?Extension: How many ways can 124 be written as the difference of two unit fractions?

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