“Schinzel's Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Schinzel's Theorem

释义

Schinzel's Theorem

For every Positive Integer , there exists a Circle in the plane having exactly LatticePoints on its Circumference. The theorem is based on the number of integral solutions to the equation

(1)

given by

(2)

where

is the number of divisors of

of the form

and

is the number of divisors of the form

. Itexplicitly identifies such circles (the Schinzel Circles) as

(3)

Note, however, that these solutions do not necessarily have the smallest possible radius.

See also Browkin's Theorem, Kulikowski's Theorem, Schinzel Circle

References

Honsberger, R. ``Circles, Squares, and Lattice Points.'' Ch. 11 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 117-127, 1973.

Kulikowski, T. ``Sur l'existence d'une sphère passant par un nombre donné aux coordonnées entières.'' L'Enseignement Math. Ser. 2 5, 89-90, 1959.

Schinzel, A. ``Sur l'existence d'un cercle passant par un nombre donné de points aux coordonnées entières.'' L'Enseignement Math. Ser. 2 4, 71-72, 1958.

Sierpinski, W. ``Sur quelques problèmes concernant les points aux coordonnées entières.'' L'Enseignement Math. Ser. 2 4, 25-31, 1958.

Sierpinski, W. ``Sur un problème de H. Steinhaus concernant les ensembles de points sur le plan.'' Fund. Math. 46, 191-194, 1959.

Sierpinski, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964.

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