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%I #49 Aug 21 2023 11:34:20
%S 3,1,4,6,2,6,4,3,6,9,9,4,1,9,7,2,3,4,2,3,2,9,1,3,5,0,6,5,7,1,5,5,7,0,
%T 4,4,5,5,1,2,4,7,7,1,2,9,1,8,7,3,2,8,7,0,1,2,3,2,4,8,6,7,1,7,4,4,2,6,
%U 6,5,4,9,5,3,7,0,9,0,7,0,7,5,9,3,1,5,3,3,7,2,1,0,8,4,8,9,0,1,4
%N Decimal expansion of sqrt(2) + sqrt(3).
%C From _Alexander R. Povolotsky_, Mar 04 2008: (Start)
%C 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.
%C _Olivier Gérard_ finds that x is 0.00343476569746030039595770020414255107204742044644777... (End)
%C Another approximation to Pi is (203*sqrt(2)+ 197*sqrt(3))/200 = 3.1414968... - _Alexander R. Povolotsky_, Mar 22 2008
%C Shape of a sqrt(8)-extension rectangle; see A188640. - _Clark Kimberling_, Apr 13 2011
%C 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
%C An algebraic integer of degree 4: largest root of x^4 - 10*x^2 + 1. - _Charles R Greathouse IV_, Sep 13 2013
%C 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
%D Emile Borel, Space and Time (1926).
%D 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.]
%D Ivan Niven, Numbers: Rational and Irrational. New York: Random House for Yale University (1961): 44.
%D Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): 44.
%H G. C. Greubel, <a href="/A135611/b135611.txt">Table of n, a(n) for n = 1..2500</a>
%H Burkard Polster, <a href="https://www.youtube.com/watch?v=D6AFxJdJYW4">Irrational roots</a>, Mathologer video (2018)
%H Karl Popper, <a href="http://web.archive.org/web/20141006082529/http://www.inf.fu-berlin.de/lehre/WS06/pmo/eng/Popper-OpenSociety.pdf">The Open Society and Its Enemies</a>, 1962.
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F Sqrt(2)+sqrt(3) = sqrt(5+2*sqrt(6)). [Landau, p. 85] - _N. J. A. Sloane_, Aug 27 2018
%e 3.14626436994197234232913506571557044551247712918732870...
%p evalf(add(sqrt(ithprime(i)), i=1..2), 118); # _Alois P. Heinz_, Jun 13 2022
%t r = 8^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
%t N[t, 130]
%t RealDigits[N[t, 130]][[1]] (* A135611 *)
%t ContinuedFraction[t, 120] (* A089078 *)
%t RealDigits[Sqrt[2] + Sqrt[3], 10, 100][[1]] (* _G. C. Greubel_, Oct 22 2016 *)
%o (PARI) sqrt(2)+sqrt(3) \\ _Charles R Greathouse IV_, Sep 13 2013
%o (Magma) SetDefaultRealField(RealField(100)); Sqrt(2) + Sqrt(3); // _G. C. Greubel_, Nov 20 2018
%o (Sage) numerical_approx(sqrt(2)+sqrt(3), digits=100) # _G. C. Greubel_, Nov 20 2018
%Y Cf. A188640, A089078, A002193, A002194, A010464.
%K nonn,cons
%O 1,1
%A _N. J. A. Sloane_, Mar 03 2008
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