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A135611
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Decimal expansion of sqrt(2) + sqrt(3).
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9
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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, 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, 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
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OFFSET
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1,1
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COMMENTS
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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
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
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
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REFERENCES
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Emile Borel, Space and Time (1926).
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.]
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.
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LINKS
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FORMULA
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Sqrt(2)+sqrt(3) = sqrt(5+2*sqrt(6)). [Landau, p. 85] - N. J. A. Sloane, Aug 27 2018
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EXAMPLE
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3.14626436994197234232913506571557044551247712918732870...
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MAPLE
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evalf(add(sqrt(ithprime(i)), i=1..2), 118); # Alois P. Heinz, Jun 13 2022
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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