This appears to be the arXiv.org link to the paper in question, "A framework of Rogers-Ramanujan identities and their arithmetic properties."<p><a href="http://arxiv.org/abs/1401.7718" rel="nofollow">http://arxiv.org/abs/1401.7718</a><p>(Submitted on 30 Jan 2014 (v1), last revised 10 Mar 2014 (this version, v2))<p>I'm surprised that there is no other discussion of this in an actual news publication about mathematics (I read those for my occupation, and am a member of several online groups that discuss mathematical research), so I wonder if the recycled press release submitted here is really the only interest that this paper has gathered in the mathematical community.
"Although no other algebraic units are as famous as the golden ratio, they are of central importance to algebra."<p>Arguably more famous algebraic numbers include:
0, 1, The square root of two, the square root of any integer, i, any integer, the nth root of any integer,...
From wikipedia on algebraic numbers:<p>> "In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalently—by clearing denominators—with integer coefficients)."<p>I almost forgot, the set of algebraic numbers also includes complex numbers.
Regarding the golden ratio ....<p>The nth number in Fibonacci sequence is denoted by f(n)<p>The golden ratio is equal to the limit of f(n)/f(n-1) as n increases without limit.<p>A recursive implementation of the nth number in the Fibonacci sequence in C:<p><pre><code> unsigned long long fib(int a) {
return a>1 ? fib(a-1)+fib(a-2) : 1;
}
</code></pre>
The above function can take a while to execute if a is sufficiently large, on my machine fib(40) takes about a second to return a value.<p>Time taken to execute the function fib(a) is denoted by t(a).<p>As n increases without limit t(n)/t(n-1) approaches the golden ratio.<p>In practise this should give you a rough value of the golden ratio.<p>The reason this happens is because the function can only increase the return value by unity, one call at a time.