EXTENDING A PUTNAM PROBLEM TO FIELDS OF VARIOUS CHARACTERISTICS
Keywords:
additive number theory, linear algebra, primitive roots, finite fields, first-order logicDOI:
https://doi.org/10.17654/0972555522037Abstract
We prove a generalization of a Putnam exam problem, to the effect that for $m \geq 2, n, r \geq 1$, and $N=m n+r$, if $x_1, x_2, \ldots, x_N$ are elements of a field $L$ of characteristic zero with the property that no matter which $r$ of the $x_i$ 's is removed, the remaining $m n$ elements can be split into $m$ groups of size $n$ with equal sums, then $x_1=x_2=\cdots=x_N$. The proof involves the Axiom of Choice. We then use primitive roots to provide a class of interesting and combinatorially meaningful counterexamples over finite prime fields. Lastly, as an elegant counterpoint to these counterexamples, we provide a constructive proof that in the special case $m=2, r=1$, the corresponding statement is valid in every field of large enough characteristic, where an effective bound is derived.
Received: October 9, 2022
Revised: November 5, 2022
Accepted: November 8, 2022
References
34th Putnam 1973, in John Scholes’s compiled list of Math problems. https://prase.cz/kalva/putnam/putn73.html (last time accessed November 9, 2022).
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Ömer Egecioglu, A combinatorial generalization of a Putnam problem, Amer. Math. Monthly 99(3) (1992), 256-258.
David J. H. Garling, Inequalities: A Journey into Linear Analysis, Cambridge University Press, Cambridge, New York, 2007.
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George F. McNulty, Elementary model theory, University of South Carolina Lecture Notes, 2016. Available online at
https://people.math.sc.edu/mcnulty/762/modeltheory.pdf.
Svetoslav Savchev and Titu Andreescu, Mathematical miniatures, Anneli Lax New Mathematical Library, 43, American Mathematical Society, 2003.
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