## MathematicsMy mathematical interests lie in what might be called 'Applied Elementary Number Theory'. For me, this means using ideas from elementary number theory to try and say something about very simple real problems, i.e. where you can actually look at a load of numbers for inspiration, or to check that your results are plausible. My main interests in this subject are Pythagorean Triples and Fibonacci Sequences, with more detail given below. ## Pythagorean TriplesPythagorean Triples such as {3, 4, 5} are very well-known but have many properties that were new to us. To simplify things, we often classify the three sides as being the odd and the even sides along with the hypotenuse, so that the triple can more conveniently be written as {e, d, h}. If the sides {e, d, h} are co-prime (which is the same as being pair-wise co-prime because of Pythagoras' Theorem), then the triple is said to be a Primitive Pythagorean Triple (or PPT). I first got interested in them after a school investigation as part of my GCSE preparations
but I have a learnt an awful lot since then! Perhaps the most important piece of information for understanding the articles below is that the sides of a PPT can be generated by
One of the things that we thought we had discovered but turned out to be well known was that the product of the three sides of any Pythagorean Triple must be divisible by 60. A proof of this can be found in many articles (and is actually easy if you use the parameterisation in terms of x and y given above) but the only online place that I have found it is here. In fact it seems that a number of the results above may be known. As is often the case it seems that we may have been Re-Searching rather than Researching. If nothing else then we hope that we bring these results to more people's attention, as many of them are beautiful in their own right. ## Fibonacci NumbersThe Fibonacci sequence is defined to be the sequence you get when you start with 0, 1 and then add the two preceding terms together to get the next one - it starts 0, 1, 1, 2, 3, 5, 8, 13 and so on. Leonardo of Pisa (a.k.a. Fibonacci) discovered the sequence in the 12th Century, but it still inspires lots of interest and there is even a serious journal (The Fibonacci Quarterly) devoted to its study. The interested reader should look no further than Ron Knott's web page for an excellent summary of all you ever wanted to know about the Fibonacci numbers. Our interest in the subject is devoted to the study of the Fibonacci sequence when it is reduced modulo a prime, p. These reduced sequences are periodic and it is not in general known what the length of the period, denoted by C(p), is. For example, reduced modulo 3, the Fibonacci sequence reads 0, 1, 1, 2, 0, 2, 2, 1, 0 so that C(3) = 8. Some general results have been obtained, most notably by Wall, to determine upper bounds for C(p) for general p, but our work gave the first conditions under which this upper bound must be attained. We also generalized some of Wall's results to the generalised Fibonacci sequence defined by the recurrence relation G
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