![]() There has been a recent breakthrough in this problem.Ģ2) Diophantine application: Cole numbersĢ3) Perfect Numbers: Perfect numbers are the sum of their factors (apart from the last factor). The twin prime conjecture states that there are infinitely many consecutive primes ( eg. It involves understanding the modulo operation.ġ8) Fermat’s last theorem: A problem that puzzled mathematicians for centuries – and one that has only recently been solved.ġ9) Natural logarithms of complex numbersĢ0) Twin primes problem: The question as to whether there are patterns in the primes has fascinated mathematicians for centuries. This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. Can all fractions with a numerator of 2 be written as 2 Egyptian fractions?ġ6) 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.ġ7) Chinese remainder theorem. ![]() Why do magic squares work?ġ4) Egyptian fractions: Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns. 3,4,5 triangle).ġ1) Mersenne primes: These are primes that can be written as 2^n -1.ġ2) Magic squares and cubes: Investigate magic tricks that use 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.ġ0) Pythagorean triples: A great introduction into number theory – investigating the solutions of Pythagoras’ Theorem which are integers (eg. The great Indian mathematician Ramanujan discovered some amazing examples of these.ħ) Patterns in Pascal’s triangle: There are a large number of patterns to discover – including the Fibonacci sequence.Ĩ) Finding prime numbers: The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics. Fermat’s Last Theorem is one of the most famous such equations.Ħ) Continued fractions: These are fractions which continue to infinity. For example, Mod 3 means the remainder when dividing by 3.Ģ) 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.Ĥ) Applications of complex numbers: The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers.ĥ) Diophantine equations: These are polynomials which have integer solutions. Suitable for Applications and Interpretations students (SL and HL) and also Analysis and Approaches students (SL and HL).ġ) Modular arithmetic – This technique is used throughout Number Theory.
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