Examples of Diophantine sets

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< Matiyasevich theorem
Yuri Vladimirovich Matiyasevich (2008), Scholarpedia, 3(7):7095. revision #42338 [link to/cite this article]

Curator: Dr. Yuri Vladimirovich Matiyasevich, Steklov Institute of Mathematics (POMI), St.Petersburg, RUSSIA

Here are some simple examples of Diophantine sets.

  • The set of all even non-negative integers is defined by the Diohpantine equation
a-2x=0
  • The set of all full squares is defined by the Diohpantine equation
a-x^2=0
  • The set of all non-negative integers that are not full squares is defined by the Pell's equation
(x+1)^2-a(y+1)^2=1

provided that the unknowns x and y range over non-negative integers.

(x^2-ax-a^2)^2=1

Suggested by: Mr. Nicolau Leal Werneck, USP, São Paulo, Brazil
Invited by: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Action editor: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Reviewer A: Dr. Alexandra Shlapentokh, Department of Mathematics, East Carolina University, Greenville, NC
Reviewer B: Dr. Martin Davis, New York University, NY
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