OFFSET
1,2
COMMENTS
This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète (see the Vieta link) using trigonometry. See the Havil reference, problem 1 (for a correction see below), pp. 69-74, and the Maor reference for Viète's approach, pp. 58-60.
The problem involves the monic Chebyshev polynomial of the first kind R(45, x) (R coefficients are given in A127672). The present problem was stated as R(45, x) = sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))) for x = sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(3))))) (see A302712). This is equivalent to R(45, 2*sin(Pi/96)) = 2*sin(15*Pi/32). It is a special case of the well known identity R(2*k+1, x) = x*(-1)^k*S(2*k, sqrt(4-x^2)), with the Chebyshev S polynomials (see A049310 for the coefficients). Take k = 22, x = 2*sin(Pi/96), and see the Havil reference, p. 71, for the proof of 2*sin(15*Pi/32) = sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))). [In the Havil reference on p. 69, the second to last exponent is 43 (not 41), and in the first problem, for the argument x a further +sqrt(2... is missing. In the general identity given on p. 71 a sign factor is missing. It should read, with n = 2*k+1: P_{2*k+1}(2*sin(theta)) = 2*(-1)^k*sin((2*k+1)*theta).]
For the argument x = sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(3))))) = 2*sin(Pi/96) = 0.65438165643552284... see A302712.
R(45, x) factorizes into minimal polynomials of 2*cos(Pi/k), named C(k, x), for short, C[k], with coefficients given in A187360 as follows. R(45, x) = C[90]*C[30]*C[18]*C[10]*C[6]*C[2]. See a comment in A127672.
All 45 zeros of R(45, x), which are real, are 2*cos((2*k+1)*Pi/90), for k = 0..44. See a comment in A127672.
Viète used the iteration, written in terms of R polynomials as R(45, x) = -R(3, -R(3, R(5, x))) (from the semigroup property of Chebyshev T polynomials). See the Maor reference, pp. 58-60. - Wolfdieter Lang, May 05 2018
An algebraic integer of degree 16. - Charles R Greathouse IV, Jan 29 2022
REFERENCES
Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.
Eli Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, pp. 56-62.
LINKS
Adriano Romano Lovaniensi, Ideae Mathematicae, 1593.
Adriano Romano Lovaniensi,Ideae Mathematicae, 1593 [alternative link].
FORMULA
This constant is 2*sin(15*Pi/32) = sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))). (for a proof see Havil. p.71).
EXAMPLE
2*sin(15*Pi/32) = 1.990369453344393772489673906218959843150949737459714123...
MATHEMATICA
RealDigits[2*Sin[15 Pi/32], 10, 120][[1]] (* Harvey P. Dale, Oct 22 2019 *)
PROG
(PARI) 2*sin(15*Pi/32) \\ Charles R Greathouse IV, Jan 29 2022
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Apr 28 2018
STATUS
approved