Diophantine EquationsDate: 11/17/97 at 15:37:25 From: Robert Felgate Subject: Diaphantine equations Dear Dr. Math, We have been investigating Pythagoras' theorem, and our teacher has suggested we search the web for information about diaphantine equations. We have not been very successful and wonder if you would help us. Thank you. Robert and his Mum Date: 11/17/97 at 16:50:59 From: Doctor Rob Subject: Re: Diaphantine equations Probably it would help if you spelled the word "Diophantine" - the equations are named after Diophantus of Alexandria, a famous Greek mathematician of antiquity. A Diophantine equation is one equation in at least two variables, say x and y, whose solutions (x,y) are required to be whole numbers (integers). Some such equations have no solutions. Some have a finite number. Some have infinitely many. The example of the Pythagorean equation is one: a^2 + b^2 = c^2. If you are asked for integer solutions, this is a Diophantine equation. You already know some solutions, I am sure: (a,b,c) = (0,0,0), (0,1,1) (0,1,-1), (0,-1,1), (0,-1,-1), (1,0,1), (1,0,-1), (-1,0,1), (-1,0,-1), for example. You can see that for this equation, you can assume that a, b, and c are all positive. From that solution you can obtain others by changing signs of some or all of a, b, and c. Furthermore, you can assume a <= b, since you can take any solution and swap a and b to get another. In this case there are infinitely many solutions. Another Diophantine equation might be 5*x + 7*y = 57. This, too, has an infinite number of solutions, and they are given by x = 3 + 7*t, y = 6 - 5*t, where t is any integer. This kind of equation is studied when you take Number Theory as an advanced undergraduate or beginning graduate student. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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