OFFSET
1,6
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.
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.
EXAMPLE
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
...
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Wolfdieter Lang, Jan 28 2015
STATUS
approved