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%I #18 Apr 12 2022 11:45:59
%S 2,9,9,6,0,5,2,6,2,9,8,6,9,2,9,9,4,6,9,2,3,4,1,3,9,4,0,2,6,2,6,3,1,8,
%T 6,3,9,7,5,8,3,0,2,1,9,1,5,0,0,5,6,4,4,4,8,1,4,0,5,2,6,3,4,0,6,5,6,0,
%U 1,0,3,4,0,4,3,5,8,8,8,9,9,8,0,2,7,1,3,2,6,1,7,9,0,9,3,9,8,2,1,8,5,3,0
%N Decimal expansion of Lagrange(4) = sqrt(1517)/13.
%C For every irrational number alpha not equivalent to each of the following three numbers i) golden section A001622, ii) sqrt(2) = A002193 and iii) (5 + sqrt(221))/14 = A177841 there exist infinitely many rational numbers h/k (in lowest terms) such that |alpha - h/k| < 1/(Lagrange(4)*k^2). The constant L(4) cannot be replaced by a larger number because then the statement becomes false for, e.g., alpha = (23 + sqrt(1517))/26. Two real numbers x and y are equivalent if there exist integers p, q, r and s with |p*s - q*r| = 1 such that y = (p*x + q)/(r*x + s) (unimodular transformation). This means that the continued fractions of x and y become eventuakky identical.
%C See the references (in Havil, p. 174, equivalence classes of numbers should have been excluded).
%C The continued fraction of Lagrange(4) is [2; repeat(1, 252, 3, 1012, 3, 252, 1, 4)]. 1/L(4) = 0.3337725078... < 1/3.
%C Perron's numbers M(xi) (pp. 4, 5), for M(xi) < 3, are the Lagrange numbers sqrt(9*Q^2 - 4)/Q, with Q = Q(n) = A002559(n), n >= 1, and his corresponding xi(4) = (sqrt(1517) + 23)/26 with a purely periodic simple continued fraction [repeat(2, 2, 1, 1, 1, 1)].
%C Cassels (p. 18) uses the version: For irrational theta not equivalent to the above given three numbers i), ii) and iii) there are infinitely many solutions of q*||q*theta|| < 1/Lagrange(4), where 1/Lagrange(4) cannot be improved for theta equivalent to -29/26 + (1/26)*sqrt(1517). Here ||x|| is the positive difference between x and the nearest integer.
%D J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.
%D Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 174-175 and 221-224.
%D J. F. Koksma, Diophantische Approximationen, 1936, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vierter Band, Teil 4, Julius Springer, Berlin, pp. 29-33.
%D Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, p. 14.
%D Oskar Perron, Über die Approximation irrationaler Zahlen durch rationale, 4. Abhandlung, pp. 1- 17, and part II., 8. Abhandlung, pp. 1-12. Sitzungsber. Heidelberger Akademie der Wiss., 1921, Carls Winters Universitätsbuchhandlung.
%D Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 10. 6 B, pp. 312-314.
%D Jörn Steuding, Diophantine Analysis, 2005, Chapman & Hall/CRC, pp. 80-82.
%H Encyclopedia of Mathematics, <a href="https://www.encyclopediaofmath.org/index.php/Diophantine_approximations">Diophantine approximations</a>.
%H A. A. Markoff, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002245841">Sur les formes quadratiques binaires indéfinies</a>, Math. Ann, 15 (18) (1879) 381-406.
%H A. A. Markoff, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002245078">Sur les formes quadratiques binaires indéfinies (Second mémoire)</a>, Math. Ann, 17 (1880) 379-399.
%F Lagrange(4) = sqrt(9*M(4)^2 - 4)/M(4) = sqrt(9*13^2 - 4)/13 = sqrt(1517)/13, with the Markoff number M(4) = A002559(4) = 13.
%e 2.9960526298692994692341394026263186397583021915005644481405263406560103404...
%t RealDigits[Sqrt[1517]/13,10,120][[1]] (* _Harvey P. Dale_, Apr 12 2022 *)
%Y The Lagrange numbers for n = 1..3 are A002163, A010466, A200991.
%Y Cf. A001622, A002559.
%K nonn,cons
%O 1,1
%A _Wolfdieter Lang_, Jun 25 2018
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