## Arctangent Diophantine EquationsApril 9, 2012 The natural numbers, which are thecounting numbers (one, two, three... and first, second, third...), are really useful things. The number, 0.25, can be expressed as 1/4; or, 2/10 + 5/100. You can see that any real number can be expressed in this way, although some numbers will be harder to express than others, and pi will take a little time.
The mathematician, Leopold Kronecker was a fan of the natural numbers. Kronecker said, "God made the integers, all the rest is the work of man." This is a mathematical version of Einstein's statement that God doesn't play dice.
There are equations called Diophantine equations that require integer solutions. I mentioned Diophantine equations in a previous article (The Lonely Runner Conjecture, August 11, 2011). The need for integer solutions to certain problems is illustrated by the cattle problem, a Diophantine problem supposedly communicated by Archimedes to his friend, Eratosthenes. You wouldn't expect a fractional cow to be grazing on the plain. A spherical cow might be a special case.
A recent article on the arXiv Preprint Server examines some trigonometric Diophantine equations.[1] This paper, by Konstantine Zelator, a mathematician at Bloomsburg University of Pennsylvania, examines equations similar to the following:
^{16}, but it isn't mathematical certainty.
## References:- Konstantine Zelator, "The Diophantine Equation arctan(1/x)+arctan(m/y)= arctan(1/k)," arXiv Preprint Server, March 28,2012
- Hasan Unal, "Proof without words: an arctangent equality," Mathematics and Computer Education, vol. 45, no. 3 (Fall 2011), p 197.
Linked Keywords: Natural numbers; real number; pi; mathematician; Leopold Kronecker; God made the integers, all the rest is the work of man; Einstein; God doesn't play dice; Diophantine equation; integer; Archimedes cattle problem; Archimedes; Eratosthenes; cattle; cow; spherical cow; arXiv Preprint Server; trigonometry; trigonometric; Bloomsburg University of Pennsylvania; equation; geometry; geometrical; proof; information wants to be free; Inkscape; triangle; angle; tangent; right triangle; similar triangles; Pythagorean theorem; high school student; computer model; source code; arctan.c; arithmetical precision; Gnumeric; Q.E.D.; mathematical certainty; fraction; numerator. |
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