Pythagoras's constant is the name used in this work for the square root of 2,
 |
(1)
|
(OEIS A002193), which the Pythagoreans proved to be irrational (Conway and Guy 1996, pp. 183-186).
The Old Babylonian tablet YBC 7289 records a sexagesimal approximation to 
on a square and its diagonal
(History of Mathematics Project).
In particular, 
is the length of the hypotenuse of an isosceles
right triangle with legs of length one. The statement that 
is irrational means
that it cannot be expressed as a ratio 
of integers 
and 
. Legend has it that the Pythagorean
philosopher Hippasus used geometric methods to demonstrate the irrationality of 
while at sea and, upon notifying
his comrades of his great discovery, was immediately thrown overboard by the fanatic
Pythagoreans. A slight generalization is sometimes known as Pythagoras's
theorem.
Theodorus subsequently proved that the square roots of the numbers from 3 to 17 (excluding 4, 9, and 16) are also irrational (Wells 1986, p. 34).
It is not known if Pythagoras's constant is normal to any base (Stoneham 1970, Bailey and Crandall 2002, p. 527).
The continued fraction for 
is periodic, as are all quadratic
surds,
![sqrt(2)=[1,2,2,2,...]=[1,2^_]](/images/equations/PythagorassConstant/NumberedEquation2.svg) |
(2)
|
(OEIS A040000).
has the Engel expansion 1, 3, 5, 5, 16, 18, 78,
102, 120, ... (OEIS A028254).
It is apparently not known if any BBP-type formula exists for 
, but 
has the formulas
 |  |  |
(3)
| ||
 |  |  |
(4)
| ||
 |  |  |
(5)
| ||
 |  |  |
(6)
| ||
 |  |  |  |  |
(7)
|
 |  |  |
(8)
|
(E. W. Weisstein, Aug. 30, 2008).
The binary representation for 
is given by
 |
(9)
|
(OEIS A004539; Graham and Pollak 1970, pp. 143-145; Bailey et al. 2004).
Using the Bhaskara-Brouncker square root algorithm for the case 
, this gives the convergents
to 
as 1, 3/2, 7/5, 17/12, 41/29, 99/70, ... (OEIS A001333
and A000129; Wells 1986, p. 34; Flannery
and Flannery 2000, p. 132; Derbyshire 2004, p. 16). The numerators are
given by the solutions to the linear recurrence
equation
 |
(10)
|
with 
and 
. Its solution is
![a(n)=1/2[(1-sqrt(2))^n+(1+sqrt(2))^n],](/images/equations/PythagorassConstant/NumberedEquation5.svg) |
(11)
|
and the denominators are the Pell numbers, i.e., solutions to the same linear recurrence equation
with 
and 
, which has solution
 |
(12)
|
Every other value of 
, i.e., 1, 7, 41, 239, ...
(OEIS A002315) produces the NSW
numbers.
Ribenboim (1996, p. 369) considers odd prime values of 
such that 
is prime. These 
are NSW numbers. The first
few such 
are 3, 5, 7, 19, 29, 47, 59, 163, 257, 421, 937, 947, 1493, 1901, ... (OEIS A005850).
For 
, the Newton's
iteration square root algorithm gives
the convergents 1, 3/2, 17/12, 577/408, 665857/470832,
... (OEIS A001601 and A051009).
The Babylonians gave the impressive approximation
 |
(13)
|
(OEIS A070197; Wells 1986, p. 35; Guy 1990; Conway and Guy 1996, pp. 181-182; Flannery 2006, pp. 32-33).
." J. Roy. Statist. Soc. Ser. A 130,
102-107, 1967.Good, I. J. and Gover, T. N. "Corrigendum."
J. Roy. Statist. Soc. Ser. A 131, 434, 1968.Gourdon, X.
and Sebah, P. "Pythagore's Constant: 
." 
."
Math. Mag. 43, 143-145, 1970. 
, and Distribution of Digits in 
and 
." Proc. Nat. Acad. Sci. U.S.A. 37,
63-67, 1951.Wells, D.