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%I #6 Oct 01 2013 17:57:40
%S 9,3,1,4,8,4,2,8,6,7,0,8,0,4,4,3,8,1,7,6,8,6,4,9,9,5,3,6,3,6,1,3,7,9,
%T 3,4,1,7,1,0,8,0,2,2,1,8,2,8,3,7,2,3,1,0,2,4,4,4,6,6,6,7,2,5,9,0,2,3,
%U 2,5,2,2,7,1,6,8,7,3,3,0,8,8,0,8,1,9,1,6,5,4,2,8,3,5,4,3,9,8,0,5
%N See comments lines for definition.
%C K = 2 in the script below. Conjecture: this diagonal expressed as a decimal is irrational and transcendental. Proof? Counterexample?
%C Write down the floating point constants x(m)>0 which solve x^2+mx=2, one per row for m=1,2,3,...:
%C 0.99999999999999999999...
%C 0.73205080756887729353...
%C 0.56155281280883027491...
%C 0.44948974278317809820...
%C 0.37228132326901432993...
%C 0.31662479035539984911...
%C and read this diagonally, the first digit after the dot from the first constant, the 2nd digit after the dot from the 2nd constant, the 3rd digit after the dot from the 3rd constant etc.
%F Also the decimal expansion of the positive solutions x of the quadratic equation x^2 + mx - 2 = 0, m = 1, 2... x = (sqrt(m^2+8)-2)/2 m=1, 2..
%o (PARI) diagonal(n,k) = { default(realprecision,n); for(m=1,n, s=.1; for(x=1,n, s=k/(s+m); ); a = Vec(Str(s)); print1(eval(a[m+2])","); ) }
%K easy,base,nonn
%O 1,1
%A _Cino Hilliard_, Jul 06 2003
%E Edited by _R. J. Mathar_, Feb 01 2008
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