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Quantizing Pythagorean triples

Hugo Mathevet, Sophie Morier-Genoud, Valentin Ovsienko

Abstract

We introduce a $q$-deformation of the Pythagoras equation $a^2 + b^2 = c^2$, which is a polynomial version of it different from the standard one. We construct a polynomial analogue, or ``$q$-analogue'', of every primitive Pythagorean triple. We also construct such analogue for a larger class of Pythagorean triples called standard. Our approach is based on the notion of $q$-deformed rational numbers and the modular group $\mathrm{PSL}(2,\mathbb{Z})$.

Quantizing Pythagorean triples

Abstract

We introduce a -deformation of the Pythagoras equation , which is a polynomial version of it different from the standard one. We construct a polynomial analogue, or ``-analogue'', of every primitive Pythagorean triple. We also construct such analogue for a larger class of Pythagorean triples called standard. Our approach is based on the notion of -deformed rational numbers and the modular group .
Paper Structure (19 sections, 5 theorems, 57 equations)

This paper contains 19 sections, 5 theorems, 57 equations.

Table of Contents

  1. The $q$-deformed Pythagoras equation
  2. An interesting class of solutions
  3. Classical Pythagorean triples
  4. $\mathrm{SL}(2,\mathbb{Z})$-action
  5. The Pythagorean tree
  6. Euclide's formula in a matrix form
  7. Relation to continued fractions
  8. A brief account on $q$-rationals
  9. An intrinsic definition
  10. The $q$-deformed $\mathrm{PSL}(2,\mathbb{Z})$-action
  11. An explicit formula
  12. The total positivity property
  13. The inverse $q$-rational
  14. A construction of $q$-Pythagorean triples
  15. The main definition
  16. ...and 4 more sections

Key Result

Theorem 1

For every standard Pythagorean triple $(a,b,c)$ there exists a solution $(\mathcal{A},\mathcal{B},\mathcal{C})$ to PythEq satisfying the conditions Con1, Con2, and Con3 and corresponding to $(a,b,c)$.

Theorems & Definitions (16)

  • Theorem 1
  • Conjecture
  • Remark
  • Example 4.1
  • Corollary 4.2
  • proof
  • Remark
  • Remark
  • Proposition 5.1
  • proof
  • ...and 6 more