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A problem in Pythagorean arithmetic. (English) Zbl 1404.03033

The paper under review can be considered as a contribution to reverse mathematics, taken in a broad sense. The statement investigated is a problem from the International Mathematical Olympiad (IMO), namely that all triples \((a,b,c)\in\mathbb{N}^{3}\) such that \(ab-c\), \(bc-a\) and \(ca-b\) are all powers of \(2\) are permutations of one of the triples \((2,2,2)\), \((2,2,3)\), \((2,6,11)\) and \((3,5,7)\). It is pointed out that, according to the standards of mathematical Olympiads, the problem needs to be solvable by “elementary means”, a claim that the paper investigates with the tools of mathematical logic.
In a first step, a subtheory of Peano arithmetic, named Pythagorean arithmetic, is given and it is argued that it captures arithmetic as done by the first of the ancient Greek mathematicians; thus, Pythagorean arithmetic arguably represents “elementary” means in a rather strong sense. Then, it is demonstrated that the problem in question is solvable in Pythagorean arithmetic. Finally, a model is constructed to show that natural fragments of Pythagorean arithmetic do not suffice for this task. As the relevant subtheories are very weak, the model can be and is given very explictly.
The paper is accessible to anyone with some background in elementary number theory and algebra.

MSC:

03C62 Models of arithmetic and set theory
11U09 Model theory (number-theoretic aspects)
11A99 Elementary number theory
03B30 Foundations of classical theories (including reverse mathematics)
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References:

[1] Kaye, R., Models of Peano Arithmetic, vol. 15 of Oxford Logic Guides, Oxford University Press, Oxford, 1991. · Zbl 0744.03037
[2] Menn S., and V. Pambuccian, “Addenda et corrigenda to ‘The arithmetic of the even and the odd’,” Review of Symbolic Logic, vol. 9 (2016), pp. 638-40. · Zbl 1382.03080
[3] Pambuccian, V. “The arithmetic of the even and the odd,” Review of Symbolic Logic, vol. 9 (2016), pp. 359-69. · Zbl 1381.03041
[4] Schacht, C. “Another arithmetic of the even and the odd,” Review of Symbolic Logic, submitted. · Zbl 1439.03103
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