A253809 Array of pairs (x,y) of Markoff triples (x,y,z) with x <= y <= z, for z given in A002559.

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

Offset: 1

Author

Wolfdieter Lang, Jan 28 2015

Keywords

• nonn
• tabf

Comments

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.

Examples

			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
...
		

References

• R. A. Mollin, Advanced Number Theory with Applications, Chapman & Hall/CRC, Boca Raton, 2010, 123-125.
• See also A002559.

Links

• Feng-Juan Chen and Yong-Gao Chen, On the Frobenius conjecture for Markoff numbers, J. Number Theory 133 (2013) 2363-2373.
• Don Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation 39:160 (1982), pp. 709-723.
• See also A002559.

Crossrefs

Cf. A002559.