A382497 Decimal expansion of 3*log(x0)/(log(8*x0/3) - 8 + Pi/sqrt(3)), where x0 is the unique real root of 96*x^3 - 786663*x^2 + 17288*x - 96 = 0.

7, 1, 0, 3, 2, 0, 5, 3, 3, 4, 1, 3, 7, 0, 0, 1, 7, 2, 7, 5, 0, 5, 7, 7, 3, 4, 2, 2, 8, 1, 0, 3, 0, 8, 4, 9, 8, 5, 2, 4, 7, 8, 9, 9, 9, 1, 7, 8, 7, 1, 8, 0, 8, 3, 3, 7, 8, 1, 3, 9, 9, 7, 1, 7, 9, 7, 3, 1, 3, 5, 8, 9, 5, 2, 1, 4, 6, 4, 6, 1, 0, 5, 9, 9, 6, 4, 2, 2, 1, 1

Offset: 1

Author

Jianing Song, Mar 29 2025

Keywords

• nonn
• cons

Comments

It is proved that the irrationality measure of Pi is at most this value. Note that for N1 and N3 defined on page 12 in the article of Zeilberger and Zudilin are given by |N1| = 8/(3*sqrt(x0)) and N3 = 8*x0/3.

Examples

			x0 = 8194.38427358233563630075...
mu(Pi) <= 3*log(x0)/(log(8*x0/3) - 8 + Pi/sqrt(3)) = 7.10320533413700172750...
		

Links

• Li Lai, Johannes Sprang, and Wadim Zudilin, A note on the irrationality of zeta_2(5), arXiv:2505.05005 [math.NT], 2025. See p. 2.
• Eric Weisstein's World of Mathematics, Irrationality Measure.
• Wikipedia, Irrationality measure.
• Doron Zeilberger and Wadim Zudilin, The irrationality measure of π is at most 7.103205334137..., arXiv:1912.06345 [math.NT], 2019-2020; Moscow Journal of Combinatorics and Number Theory, 9 (4): 407-419.

Crossrefs

Cf. A000796, A002194, A093602.

Programs

PARI

my(x0 = solve(x=8194, 8195, 96*x^3 - 786663*x^2 + 17288*x - 96)); 3*log(x0)/(log(8*x0/3) - 8 + Pi/sqrt(3))