Theodorus's Constant

There are (at least) two mathematical constants associated with Theodorus. The first Theodorus's constant is the elementary algebraic number , i.e., the square root of 3. It has decimal expansion

(1)

(OEIS A002194) and is named after Theodorus, who proved that the square roots of the integers from 3 to 17 (excluding squares 4, 9,and 16) are irrational (Wells 1986, p. 34). The space diagonal of a unit cube has length .

 has continued fraction [1, 1, 2, 1, 2, 1, 2, ...] (OEIS A040001). In binary, it is represented by

(2)

(OEIS A004547).

Another constant sometimes known as the constant of Theodorus is the slope of a continuous analog of the discrete Theodorus spiral due to Davis (1993) at the point , given by

			(3)
			(4)
			(5)
			(6)

(OEIS A226317; Finch 2009), where  is the Riemann zeta function.

REFERENCES:

Davis, P. J. Spirals from Theodorus to Chaos. Wellesley, MA: A K Peters, 1993.

Finch, S. "Constant of Theodorus." https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.440.3922&rep=rep1&type=pdf.

Gautschi, W. "The Spiral of Theodorus, Numerical Analysis, and Special Functions." https://www.cs.purdue.edu/homes/wxg/slidesTheodorus.pdf.

Jones, M. F. "Approximations to the Square Roots of the Primes Less Than 100." Math. Comput. 22, 234-235, 1968.

Sloane, N. J. A. Sequences A002194/M4326, A004547, A040001, and A226317 in "The On-Line Encyclopedia of Integer Sequences."

Uhler, H. S. "Approximations Exceeding  Decimals for , , , and Distribution of Digits in Them." Proc. Nat. Acad. Sci. USA 37, 443-447, 1951.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 34-35, 1986.