Student Reports: A Rewarding Undertaking - Undergraduate Report (cont.)

Diophantine Equations

Diophantus, often known as the “father of algebra,” was a Greek mathematician in approximately 250 AD. He was the first to attempt algebraic notation. In his Arithmetica, a work on solutions of algebraic equations and the theory of numbers, he showed how to solve simple and quadratic equations. Diophantus was interested in exact solutions rather than approximate solutions considered perfectly appropriate. Diophantus found interest in polynomial equations in one or more variables for which it is necessary to find a solution in integer form. He found solutions to equations that are negative or irrational square roots to be useless.

Diophantine Equation - An equation in which the coefficients are integers and the solutions are also required to be integers.

Examples of Diophantine Equations:

x + y = 5          linear

x²+7=2ⁿ

(2x–1)²=2ⁿ –7

x² + p = 2ⁿ

x³ + 3 = 4ⁿ

xⁿ + yⁿ = zⁿ, for n = 2    Pythagoras’ Theorem,

for n ≥ 3 Fermat's Last Theorem (that there is no integer solution)

x² – dy² = 1      Pell's equation

ax + by = 1      Bezout's identity

A⁴ + B⁴ = C⁴ + D⁴