OFFSET
1,1
COMMENTS
From Alexander R. Povolotsky, Mar 04 2008: (Start)
The value of sqrt(2) + sqrt(3) ~= 3.146264369941972342329135... is "close" to Pi. [See Borel 1926. - Charles R Greathouse IV, Apr 26 2014] We can get a better approximation by solving the equation: (2-x)^(1/(2+x)) + (3-x)^(1/(2+x)) = Pi.
Olivier Gérard finds that x is 0.00343476569746030039595770020414255107204742044644777... (End)
Another approximation to Pi is (203*sqrt(2)+ 197*sqrt(3))/200 = 3.1414968... - Alexander R. Povolotsky, Mar 22 2008
Shape of a sqrt(8)-extension rectangle; see A188640. - Clark Kimberling, Apr 13 2011
This number is irrational, as instinct would indicate. Niven (1961) gives a proof of irrationality that requires first proving that sqrt(6) is irrational. - Alonso del Arte, Dec 07 2012
An algebraic integer of degree 4: largest root of x^4 - 10*x^2 + 1. - Charles R Greathouse IV, Sep 13 2013
Karl Popper considers whether this approximation to Pi might have been known to Plato, or even conjectured to be exact. - Charles R Greathouse IV, Apr 26 2014
REFERENCES
Emile Borel, Space and Time (1926).
Ivan Niven, Numbers: Rational and Irrational. New York: Random House for Yale University (1961): 44.
Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): 44.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..2500
Susan Landau, Simplification of nested radicals, SIAM Journal on Computing 21.1 (1992): 85-110. See page 85. [Do not confuse this paper with the short FOCS conference paper with the same title, which is only a few pages long.]
Burkard Polster, Irrational roots, Mathologer video (2018)
Karl Popper, The Open Society and Its Enemies, 1962.
FORMULA
Sqrt(2)+sqrt(3) = sqrt(5+2*sqrt(6)). [Landau, p. 85] - N. J. A. Sloane, Aug 27 2018
Equals 1/A340616. - Hugo Pfoertner, May 08 2024
Equals Product_{k>=0} (((4*k + 1)*(12*k + 11))/((4*k + 3)*(12*k + 1)))^(-1)^k. - Antonio Graciá Llorente, May 22 2024
EXAMPLE
3.14626436994197234232913506571557044551247712918732870...
MAPLE
evalf(add(sqrt(ithprime(i)), i=1..2), 118); # Alois P. Heinz, Jun 13 2022
MATHEMATICA
r = 8^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A135611 *)
ContinuedFraction[t, 120] (* A089078 *)
RealDigits[Sqrt[2] + Sqrt[3], 10, 100][[1]] (* G. C. Greubel, Oct 22 2016 *)
PROG
(PARI) sqrt(2)+sqrt(3) \\ Charles R Greathouse IV, Sep 13 2013
(Magma) SetDefaultRealField(RealField(100)); Sqrt(2) + Sqrt(3); // G. C. Greubel, Nov 20 2018
(Sage) numerical_approx(sqrt(2)+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane, Mar 03 2008
STATUS
approved