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A253809
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Array of pairs (x,y) of Markoff triples (x,y,z) with x <= y <= z, for z given in A002559.
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0
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1, 1, 1, 1, 1, 2, 1, 5, 2, 5, 1, 13, 1, 34, 2, 29, 5, 13, 1, 89, 5, 29, 1, 233, 2, 169, 13, 34, 1, 610, 5, 194, 1, 1597, 2, 985, 5, 433, 13, 194, 34, 89, 1, 4181, 29, 169, 1, 10946, 2, 5741, 29, 433, 5, 2897, 13, 1325, 89, 233, 1, 28657
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OFFSET
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1,6
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COMMENTS
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Frobenius' conjecture on Markoff triples is that the maximal member z of the triple of positive integers (x,y,z), satisfying x^2 + y^2 + z^2 - 3*x*y*z = 0, with x <= y <= z, determines x and y uniquely. Also, each entry from A002559 (Markoff numbers) is conjectured to appear as a maximal member z. If an entry A002559(n) should not appear as z then one puts z(n) = 0 and row n will be 0, 0.
If this Frobenius conjecture is true then the row length of this array is always 2, and only positive numbers appear.
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REFERENCES
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R. A. Mollin, Advanced Number Theory with Applications, Chapman & Hall/CRC, Boca Raton, 2010, 123-125.
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LINKS
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EXAMPLE
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The array A(n,k) begins:
If the Frobenius conjecture is true there will only be one pair x(1,n), y(1,n) for each z(n).
n z(n) \ k=1: x(1,n) k=2: y(1,n) ...
1 1: 1 1
2 2: 1 1
3 5: 1 2
4 13: 1 5
5 29: 2 5
6 34: 1 13
7 89: 1 34
8 169: 2 29
9 194: 5 13
10 233: 1 89
11 433: 5 29
12 610: 1 233
13 985: 2 169
14 1325: 13 34
15 1597: 1 610
16 2897: 5 194
17 4181: 1 1597
18 5741: 2 985
19 6466: 5 433
20 7561: 13 194
21 9077: 34 89
22 10946: 1 4181
23 14701: 29 169
24 28657: 1 10946
25 33461: 2 5741
26 37666: 29 433
27 43261: 5 2897
28 51641: 13 1325
29 62210: 89 233
30 75025: 1 28657
...
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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