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Definition: Diophantine Equations

A Diophantine equation (named after Diophantos of Alexandria) is an equation with one or more integer variables $x,y,z,\ldots$ (often involving also their powers) and with integer coefficients $a,b,c,\ldots$

A Diophantine equation is called linear quadratic, cubic, etc. or, in general, $n$th order) if all variables have at most the power of $1$ ($2$, $3$, etc. $n$).

Examples of Diophantine Equations

• $2{x} +5y=16$ (linear),
• $-x_1+2x_2+3x_3-3x_4=0$ (linear),
• $x^2+y^2=z^2$ (quadratic),
• $2xy+y^2-z^3=1$ (cubic).
• for $x^n+yz=z^2$ ($n$th order).

It is convenient to write a given Diophantine equation in the form $f(x_1,\ldots,x_r)=0$ where $x_1,\ldots,x_r$ are the variables of this equation. The above equation defines interpreted as a function $f:\mathbb Z^r\to \mathbb Z.$ The above examples can be re-written as

• $2{x} +5y-16=0$,
• $-x_1+2x_2+3x_3-3x_4=0$ (already in this form),
• $x^2+y^2-z^2=0$,
• $x^4+y^4-z^2=0$,
• $2xy+y^2-z^3-1=0$
• $x^n+yz-z^2=0.$

Mentioned in:

Proofs: 1 2 3
Propositions: 4 5 6

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References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998