My 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 Triples

Pythagorean 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 x and y which are co-prime and of opposite parity, where we then have the three relations: e = 2xy, d = x2 - y2, h = x2 + y2 and w.l.o.g. we take x > y.

TitleMain IdeaPublished?Co-worker(s)
An extension of the Fundamental Theorem on Right-Angled Triangles We show that any integer that plays the role of e, d or h in a PPT plays that same role in 2k-1 PPTs, where k is the number of distinct prime factors Math. Gaz 89 (2005), 237
PDF (188 KB)
A. Vella and J. Wolf
When is n a member of a Pythagorean Triple? Does exactly what it says on the tin! We give necessary and sufficient conditions for each of the sides in turn. Math. Gaz. 87 (2003), 102
PDF (28 KB)
A. Vella
More properties of Pythagorean Triples Here we look at some assorted properties of PPTs Math. Gaz. 85 (2001), 275
PDF (24 KB)
A. Vella

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 Numbers

The 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 Gn = aGn-1+bGn-2 with various starting values.

TitleMain IdeaPublished?Co-worker(s)
Calculating exact cycle lengths in the generalised Fibonacci sequence modulo p We consider almost the most general Fibonacci sequence and combine ideas in the previous two articles to calculate C(p) exactly in some special cases. Math. Gaz. 90 (2006), 70
PDF (116 KB)
A. Vella
Some properties of finite Fibonacci sequences We use some of the ideas in the earlier paper applied to the case a=b=1 to give the exact value of C(p) under special circumstances. Math. Gaz. 88 (2004), 494 A. Vella
Cycles in the generalised Fibonacci sequence modulo a prime This article looks at the generalized Fibonacci sequence in the special case that a2+4b is a perfect square (for technical reasons) and gave some exact values for C(p) Math. Mag. 75 (2002), 294
PDF (112 KB)
A. Vella

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This page was most recently updated on the 6th of December, 2007.