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For n = 14 residues congruent to 0, 3, or 4 mod 7 are forbidden. (See comments to A370164 for explanation.) All other residues occur. For example, the Markov numbers 1, 2, 5, 34, 610, 1325, 194, and 13 produce the residues shown in row 14 of the triangle (mod 14).

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0, 0, 1, 1, 2, 1, 2, 0, 1, 2, 3, 4, 1, 2, 4, 5, 1, 2, 5, 6, 1, 2, 5, 1, 2, 4, 5, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 4, 5, 6, 7, 9, 10, 1, 2, 5, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 5, 6, 8, 9, 12, 13, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 1, 2, 5, 9, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17

Martin Aigner, <a href="https://archive.org/details/markovstheorem100000aign.html">Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings</a>, [archive.org copy of the book].

def generateAllMarkovTreeResidues(kn):

row = [[1 % k, n, 5 % k, n, 2 % kn]]

newRow.append([trpl[0], (3*trpl[0]*trpl[1]-trpl[2]) % k, n, trpl[1]])

newRow.append([trpl[1], (3*trpl[1]*trpl[2]-trpl[0]) % k, n, trpl[2]])

sum[r for n in range(1, 16) for r in generateAllMarkovTreeResidues(k) for k in range(1, 19)n)]

Markov tree: A32745, A327345, A368546.

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Martin Aigner, <a href="https://archive.org/details/markovstheorem100000aign.html">Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings</a>, [archive.org copy of the book].

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allocated for William P. Orrick

Irregular triangle read by rows: row n is the list of residues mod n that occur among the Markov numbers

0, 0, 1, 1, 2, 1, 2, 0, 1, 2, 3, 4, 1, 2, 4, 5, 1, 2, 5, 6, 1, 2, 5, 1, 2, 4, 5, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 4, 5, 6, 7, 9, 10, 1, 2, 5, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 5, 6, 8, 9, 12, 13, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 1, 2, 5, 9, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17

1,5

Length of row n is A370164(n).

Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013. x+257 pp. ISBN: 978-3-319-00887-5; 978-3-319-00888-2 MR3098784.

Martin Aigner, <a href="https://archive.org/details/markovstheorem100000aign">Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings</a>, [archive.org copy of the book].

The first rows are:

n

1: 0

2: 0 1

3: 1 2

4: 1 2

5: 0 1 2 3 4

6: 1 2 4 5

7: 1 2 5 6

8: 1 2 5

9: 1 2 4 5 7 8

10: 0 1 2 3 4 5 6 7 8 9

11: 1 2 4 5 6 7 9 10

12: 1 2 5 10

13: 0 1 2 3 4 5 6 7 8 9 10 11 12

14: 1 2 5 6 8 9 12 13

15: 1 2 4 5 7 8 10 11 13 14

16: 1 2 5 9 13

17: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18: 1 2 4 5 7 8 10 11 13 14 16 17

19: 1 2 3 4 5 6 8 9 10 11 13 14 15 16 17 18

20: 1 2 5 6 9 10 13 14 17 18

(SageMath)

def generateAllMarkovTreeResidues(k):

row = [[1 % k, 5 % k, 2 % k]]

residuesFound = []

triplesFound = []

while row != []:

newRow = []

for trpl in row:

if trpl[1] not in residuesFound:

residuesFound.append(trpl[1])

if trpl[2] < trpl[0]:

trpl.reverse()

if trpl not in triplesFound:

triplesFound.append(trpl)

newRow.append([trpl[0], (3*trpl[0]*trpl[1]-trpl[2]) % k, trpl[1]])

newRow.append([trpl[1], (3*trpl[1]*trpl[2]-trpl[0]) % k, trpl[2]])

row = newRow

residuesFound.sort()

return(residuesFound)

sum(generateAllMarkovTreeResidues(k) for k in range(1, 19))

Markov numbers: A002559.

Markov tree: A32745, A368546.

Cf. A370164.

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Wouter Meeussen and William P. Orrick, Mar 03 2024

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