Algebra and number1) Modular arithmetic - This technique is used throughout Number Theory. For example, Mod 3 means the remainder when dividing by 3.
2) Goldbach’s conjecture: “Every even number greater than 2 can be expressed as the sum of two primes.”  One of the great unsolved problems in mathematics.
3) Probabilistic number theory
4) Applications of complex numbers: The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers.
5) Diophantine equations: These are polynomials which have integer solutions.  Fermat’s Last Theorem is one of the most famous such equations.
6) Continued fractions: These are fractions which continue to infinity.  The great Indian mathematician Ramanujan discovered some amazing examples of these.
7) Patterns in Pascal’s triangle: There are a large number of patterns to discover – including the Fibonacci sequence.
8) Finding prime numbers:  The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics.  There is a $1 million prize for solving the Riemann Hypothesis and $250,000 available for anyone who discovers a new, really big prime number.
9) Random numbers
10) Pythagorean triples: A great introduction into number theory – investigating the solutions of Pythagoras’ Theorem which are integers (eg. 3,4,5 triangle).
11) Mersenne primes: These are primes that can be written as 2^n -1.
12) Magic squares and cubes: Investigate magic tricks that use mathematics.  Why do magic squares work?
13) Loci and complex numbers
14) Egyptian fractions: Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns.  2/3 could be written as 1/6 + 1/2.  Can all fractions with a numerator of 2 be written as 2 Egyptian fractions?
15) Complex numbers and transformations
16) Euler’s identity: An equation that has been voted the most beautiful equation of all time, Euler’s identity links together 5 of the most important numbers in mathematics.
17) Chinese remainder theorem.  This is a puzzle that was posed over 1500 years ago by a Chinese mathematician.  It involves understanding the modulo operation.
18) Fermat’s last theorem: A problem that puzzled mathematicians for centuries – and one that has only recently been solved.
19) Natural logarithms of complex numbers
20) Twin primes problem: The question as to whether there are patterns in the primes has fascinated mathematicians for centuries.  The twin prime conjecture states that there are infinitely many consecutive primes ( eg. 5 and 7 are consecutive primes).  There has been a recent breakthrough in this problem.
21) Hypercomplex numbers
22) Diophantine application: Cole numbers
23) Odd perfect numbers: Perfect numbers are the sum of their factors (apart from the last factor).  ie 6 is a perfect number because 1 + 2 + 3 = 6.
24) Euclidean algorithm for GCF
25) Palindrome numbers: Palindrome numbers are the same backwards as forwards.
26) Fermat’s little theorem:  If p is a prime number then a^p – a is a multiple of p.
27) Prime number sieves
28) Recurrence expressions for phi (golden ratio): Phi appears with remarkable consistency in nature and appears to shape our understanding of beauty and symmetry.
29) The Riemann Hypothesis – one of the greatest unsolved problems in mathematics – worth $1million to anyone who solves it (not for the faint hearted!)
30) Time travel to the future: Investigate how traveling close to the speed of light allows people to travel “forward” in time relative to someone on Earth.  Why does the twin paradox work?