A305308 Decimal expansion of Lagrange(4) = sqrt(1517)/13.

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, 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, 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

Offset

1,1

Comments

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.

See the references (in Havil, p. 174, equivalence classes of numbers should have been excluded).

The continued fraction of Lagrange(4) is [2; repeat(1, 252, 3, 1012, 3, 252, 1, 4)]. 1/L(4) = 0.3337725078... < 1/3.

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)].

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.

References

J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 174-175 and 221-224.

J. F. Koksma, Diophantische Approximationen, 1936, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vierter Band, Teil 4, Julius Springer, Berlin, pp. 29-33.

Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, p. 14.

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.

Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 10. 6 B, pp. 312-314.

Jörn Steuding, Diophantine Analysis, 2005, Chapman & Hall/CRC, pp. 80-82.

Links

Table of n, a(n) for n=1..103.

Encyclopedia of Mathematics, Diophantine approximations.

A. A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann, 15 (18) (1879) 381-406.

A. A. Markoff, Sur les formes quadratiques binaires indéfinies (Second mémoire), Math. Ann, 17 (1880) 379-399.

Formula

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.

Example

2.9960526298692994692341394026263186397583021915005644481405263406560103404...

Mathematica

RealDigits[Sqrt[1517]/13, 10, 120][[1]] (* Harvey P. Dale, Apr 12 2022 *)

Crossrefs

The Lagrange numbers for n = 1..3 are A002163, A010466, A200991.

Cf. A001622, A002559.

Sequence in context: A021889 A309927 A016643 * A087042 A266274 A003678

Adjacent sequences: A305305 A305306 A305307 * A305309 A305310 A305311

Keyword

nonn,cons

Author

Wolfdieter Lang, Jun 25 2018

Status

approved