Simon> Hello, Yes, this is the argument, the length of the proposed approximation is about the length in number of digits of precision. Just a coïncidence, and there are others, very few can be explained fully. here are a few of them : http://mathworld.wolfram.com/AlmostInteger.html [Great collection--much new to me. --rwg] http://fr.wikipedia.org/wiki/Nombre_presque_entier (in french), http://en.wikipedia.org/wiki/Almost_integer ["This article may need to be rewritten entirely to comply with Wikipedia's quality standards." I agree. Its quality is too high." --rwg] Simon Plouffe On 25/08/2014 05:56, Leo Broukhis wrote: Is there a mathematical reason why the root of cos(x)=x is very close to r=exp(ln(pi/160)/13); (r and cos(r) differ by ~4x10^-7), or is it just a coincidence? GarethM><handwave>It doesn't seem like the number of bits of agreement is any bigger than the number of bits you needed to specify the number, so I expect it's just coincidence.</handwave> -- g Note that by http://www.tweedledum.com/rwg/idents.htm (d218, ~45%) the true root is given by the Kapteyn series Out[49]= 2*Sum[((-1)^k*BesselJ[1 + 2*k, 1 + 2*k])/(1 + 2*k), {k, 0, Infinity}] In[50]:= N[{%, (Pi/160)^(1/13)}] Out[50]= {0.739085133215157, 0.7390853722244995} Convergence: lousy. --rwg