Kulikowski's Theorem

A question posed by Hugo Steinhaus to secondary school teachers in the late 1950s led within short time to generalizations and elegant variations. Here, I am concerned with the latter. First, André Schinzel showed that, for any, non-negative integer $n$ there is a circle that passes through exactly $n$ integer grid points. Schinzel proof, albeit short, depended on a lemma that [R. Honsberger, p. 119] identified as rather long. A short proof of the lemma is now available online.

Thadée Kulikowski has extended Schinzel's result to higher dimensions:

Let $m$ be an integer, $m\ge 3.$ For every natural number $n,$ there exists a sphere $\displaystyle\sum_{k=1}^{m}(x_{k}-a_{k})^{2}=r^2$ in $\mathbb{R}^m$ that contains exactly $n$ grid points (i.e. points with integer coordinates $x_{k},\space k=1,2,\ldots,m.$)

Amazingly, the extension only draws on the properties of irrational numbers that were utilized in solving the original question by Steinhaus.

Proof

Thus we assume Schinzel's result: for every integer $n\ge 0$ there is a circle $(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}=c$ that passes through exactly $n$ points with integer coordinates.

There are numbers $a_k,\space k=3,\ldots,m$ such that $\sum_{k=3}^{m}c_{k}a_{k},$ for all $c_{k}$ rational, implies $c_{3}=\ldots=c_{m}=0.$ In other words, the numbers $a_k,\space k=3,\ldots,m$ are assumed to be linearly independent over $\mathbb{Q},$ the set of rational numbers. Now, consider the sphere

$\displaystyle\sum_{k=1}^{m}(x_{k}-a_{k})^{2}=c+\sum_{k=3}^{m}a_{k}^{2}.$

We'll prove that this sphere passes through exactly $n$ integer points. Indeed, from the equations of the sphere,

$\displaystyle 2\sum_{k=3}^{m}x_{k}a_{k}=(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}+\sum_{k=3}^{m}x_{k}^{2}-c=d.$

The assumption that all $x_{k}$ are integer makes $d$ rational. Thus, by the choice of $a_{k}$'s, $x_{3}=\ldots=x_{m}=0,$ implying that, as a matter of fact, point $(x_{1},\ldots,x_{m})$ lies on the circle $(x_{1}-a_{1})^{2}+(x_{2}-a_{2})^{2}=c$, which was chosen to contain $n$ such points, to start with.

References

  1. R. Honsberger, Mathematical Gems, MAA, 1973, pp. 117-127
  2. T. Kulikowski, Sur L'Existence d'Une Sphère Passant par un Nombre Donné de Points aux Coordonnées Entières, L'Enseignement Mathématique, Volume 5 (Series 2), 1959, 89-90 (available online)
  3. A. Schinzel, Sur L'Existence d'Un Cercle Passant par Un Nombre Donné de Points aux Coordonnées Entières, L'Enseignement Mathématique, Volume 4 (Series 2), 1958, 71-72 (available online)
  4. W. Sierpinski, Sur Quelques Problemes Concernant les Points aux Coordonnées Entières, L'Enseignement Mathématique, Volume 4 (Series 2), 1958, 25-31 (available online)

Pascal's Triangle and the Binomial Coefficients

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