Abstract: We prove a generalization of a Putnam exam problem, to the effect that for m ≥ 2, n, r ≥ 1, and N = mn + r, if x1, x2, ... xN are elements of a field L of characteristic zero with the property that no matter which r of the xi's is removed, the remaining mn elements can be split into m groups of size n with equal sums, then x1 = x2 = … = xN. 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.
|
Keywords and phrases: additive number theory, linear algebra, primitive roots, finite fields, first-order logic.
Received: October 9, 2022; Revised: November 5, 2022; Accepted: November 8, 2022; Published: November 17, 2022
How to cite this article: Mihai Caragiu and Rachael Harbaugh, Extending a Putnam problem to fields of various characteristics, JP Journal of Algebra, Number Theory and Applications 59 (2022), 33-45. http://dx.doi.org/10.17654/0972555522037
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License
References:
[1] 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). [2] Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, 1976. [3] Ömer Eğecioğlu, A combinatorial generalization of a Putnam problem, Amer. Math. Monthly 99(3) (1992), 256-258. [4] David J. H. Garling, Inequalities: A Journey into Linear Analysis, Cambridge University Press, Cambridge, New York, 2007. [5] Hans Hermes, Introduction to Mathematical Logic, Springer, Universitext, 1973. [6] 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. [7] Svetoslav Savchev and Titu Andreescu, Mathematical miniatures, Anneli Lax New Mathematical Library, 43, American Mathematical Society, 2003.
|