Welcome guest
You're not logged in.
139 users online, thereof 0 logged in

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),
  • $x^2+y^2-z^2=0$,
  • $x^4+y^4-z^2=0$,
  • $2xy+y^2-z^3-1=0$
  • $x^n+yz-z^2=0.$

| | | | | created: 2019-04-13 09:09:12 | modified: 2019-04-19 06:48:47 | by: bookofproofs | references: [1272], [8152]


This work was contributed under CC BY-SA 3.0 by:

This work is a derivative of:

(none)

Bibliography (further reading)

[8152] Jones G., Jones M.: “Elementary Number Theory (Undergraduate Series)”, Springer, 1998

[1272] Landau, Edmund: “Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

FeedsAcknowledgmentsTerms of UsePrivacy PolicyImprint
© 2018 Powered by BooOfProofs, All rights reserved.