Modified Farey Sequences

Transcription

1 Modified Farey Sequences Dhroova Aiylam MIT PRIMES May, Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

2 The Farey Sequence a b, c d a + c b + d Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

3 Properties Let a b, c a+c d be two consecutive fractions, with their mediant equal to b+d. Some useful properties: The mediant never needs to be reduced bc ad equals The first half of the list of numerators equals the previous Denominators are increased by the numerator Theorem (Well Known) Every rational number between and appears somewhere. Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

4 Ford Circles Begin with circles, radius, placed at (, ) and (, ) Between any two tangent circles, insert a circle tangent to the two and the axis x-coordinates are given by the Farey Sequence Curvatures equal twice denominator squared Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

5 Visual representation of irreducibility Odd denominators are red Even denominators are even Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

6 A Different Farey Sequence a b, c d a + c b + d, a + c b + d Here we insert weighted mediants. Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, 6 / 8

7 The General Farey Sequence In general, insert k fractions; weights sum to k. The case above is k =, and the original is k =. We pose the following questions: For which k do all rational numbers appear? Can we categorize which rationals appear based on k? How do the properties of the original sequence generalize? Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

8 Some lemmas for k = For consecutive fractions a b and c d : Lemma bc ad is a power of. Lemma b and d are odd Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, 8 / 8

9 Some more lemmas for k = Lemma gcd (a + c, b + d) = gcd (a + c, b + d) {, }. Lemma The one with smaller denominator is the closest rational approximation with smaller, odd denominator to the other. Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

10 A couple of theorems k = N(r, i) is numerator of i th fraction in row r. D(r, i) defined similarly. For any consecutive rows n and n + : Theorem N(n, i) = N(n +, i) Theorem N(n, i) + D(n, i) = D(n +, i) Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

11 Numerators 6 Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

12 and Denominators 6 Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

13 Determinant bc ad is called the determinant of a b, c d Notice the interesting fractal-like behavior. Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

14 The Determinant for k = The following is a visual representation of the power of three in the determinant in a row: The recursive rule found for this sequence is quite involved, and was hence omitteed. Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

15 The Determinant in General For prime k: Determinant is a power of k The list of determinants in row n is the set {, k,..., k n } In general: Determinant divides k n GCD divides determinant For k odd, every divisor of k n appears among the list of determinants in row n. Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

16 The General Case Most interesting for prime k Stronger conclusions, non-trivial reduction Not all numbers appear Theorem For odd prime k, only those numbers with denominators that are mod k can appear. For odd prime k, all rational numbers with denominators mod k appear even k, all rational numbers appear somewhere For Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, 6 / 8

17 Future Research Proving any unproven conjectures Completely finishing up the case k = Inventing a continued fraction variant Generalizing results to prime, and then all k Finding, for all k, exactly which fractions appear Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, / 8

18 Acknowledgements MIT PRIMES Tanya Khovanova (for providing mentorship and guidance) Jim Propp (for suggesting the project) Dhroova Aiylam (MIT PRIMES) Modified Farey Sequences May, 8 / 8