Diophantine Equations

Diophantine equations are so named after a revolutionary mathematician Diophantus of whom very little is known, except that he lived and worked in the city of Alexandria in the present day Egypt on the Mediterranean coast. His life span is placed somewhere between 150 BC and 300 AD. He is best known as the author of Arithmetica, on a later edition of which B. Fermat left a short note that eventually became the Last Fermat Theorem (FLT).

Diophantine equations are equations with integer coefficients in one or more variables. The solutions sought are also integers.

Diophantine equations are quite finicky: slight modifications may lead from a solvable equation to one with no solutions. Methods that work for one equation may prove useless for a very similar equation.

The FLT has no solutions for n ≥ 3. Its slight modification has an infinite family of solutions. Here is another example: the equation x + y + z = x² + y² + z² has a good deal of rational solutions

but only trivial integer ones (say, x = 1, y = z = 0.) However, the equation (x + y + z)² = x² + y² + z² infinitely many such solutions. The latter equation is equivalent to

xy + yz + zx = 0,

from where z = - xy/(x + y). Let α = gcd(x, y) and x = αu and y = αv, gcd(u, v) = 1. Then z = - αuv/(u + v). All that remains to exhibit and integer solution to xy + yz + zx = 0 one needs to choose α a multiple of u + v, say α = w(u + v). With this we get a three-parameter family of solutions:

x = w(u + v)u,
y = w(u + v)v,
z = -wuv.

Verification is immediate. This is an example of the parametric method of solution.

Another equation (M. Klamkin, Math Mag, Jan-Feb 1961, 182) is solved by factoring:

(x⁴ + y⁴ + z⁴)² = 2(x⁸ + y⁸ + z⁸).

It so happens that (do verify that)

2(x⁸ + y⁸ + z⁸) - (x⁴ + y⁴ + z⁴)² = (x² + y² + z²)(x² + y² - z²)(z² - y² + x²)(z² + y² - x²),

from which it follows that any Pythagorean triple solves that equation.

References

1. T. Andreescu, D. Andrica, I. Cucurezeanu, An Introduction to Diophantine Equations, Birkhäuser, 2010, p. 23
2. I. G. Bashmakova, Diophantus and Diophantine Equations, MAA, 1997
3. E. Burger, Exploring the Number Jungle: a Journey into Diophantine analysis, AMS, 2000

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