OFFSET
1,6
COMMENTS
The positive integers x, y, z satisfy the Diophantine equation x^2 + y^2 + z^2 = 3*x*y*z, 1 <= x <= y <= z.
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.31.3 Markov-Hurwitz Equation, p. 200.
LINKS
Eric Weisstein's World of Mathematics, Markov Number.
Wikipedia, Markov number.
EXAMPLE
The array of Markov triples begins:
(1, 1, 1),
(1, 1, 2),
(1, 2, 5),
(1, 5, 13),
(2, 5, 29),
(1, 13, 34),
...
MATHEMATICA
triples = 30; depth0 = 10 (* adjust depth according to message after first run *) ; Clear[zz, fx, fy]; fx[1] = fy[1] = fx[2] = fy[2] = fx[5] = 1;
fy[5] = 2; zz[n_] := zz[n] = Module[{f, x, y, z}, f[] = {1, 2, 5}; f[ud___, u(*up*)] := f[ud, u] = Module[{g = f[ud]}, x = g[[1]]; y = g[[3]]; z = 3*g[[1]]*g[[3]] - g[[2]]; fx[z] = x; fy[z] = y; {x, y, z}]; f[ud___, d(*down*)] := f[ud, d] = Module[{g = f[ud]}, x = g[[2]]; y = g[[3]]; z = 3*g[[2]]*g[[3]] - g[[1]]; fx[z] = x; fy[z] = y; {x, y, z}]; f @@@ Tuples[{u, d}, n] // Flatten // Union // PadRight[#, triples]&]; zz[n = depth0]; zz[n++]; While[zz[n] != zz[n - 1], n++]; Print["depth = n = ", n]; xyz = {fx[#], fy[#], #} & /@ zz[n]; Flatten[xyz]
PROG
(PARI) N=5000;
for(k=1, N, for(j=1, k, for(i=1, j, if(i*j>k, break); if(i^2+j^2+k^2==3*i*j*k, print1(i, ", ", j, ", ", k, ", "))))); \\ Seiichi Manyama, Feb 16 2022
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Jean-François Alcover, Aug 30 2017
STATUS
approved