User blog comment:Nayuta Ito/Brand-new Googologism from Algebra/@comment-11227630-20170924123530/@comment-1605058-20170925084746

See first this paper. Theorem 3.1 describes a system of 8 equations in 15 variables such that its solutions in the natural numbers with \(a>1\) have \(x=x_k(a)\). By adding equations of the form \(v=v_1^2+v_2^2+v_3^2+v_4^2\) we replace this with a problem in which variables can range over integers, but the original variables are forced to be nonnegative. Summing the squares of these equations gives a single equation \(\Phi(x,k,a)\) with (I think) 72 implicit variables which has a solution iff \(x=x_k(a)\). If we fix, say, \(a=2\), then \(x\) has to grow exponentially in \(k\). Therefore an equation, say, \((x_1-5)^2+(\Phi(x_1,5,2)-x_2)^2+(\Phi(x_2,5,2)-x_3)^2+(\Phi(x_3,5,2)-x_4)^2=0\) has only solutions in which \(x_4\) is of the order \(2^{2^{2^5}}\).

This is a rather nontrivial piece of mathematics here, so feel free to ask about details. For the record, the paper linked proves that every recursive function can be described in this way, not just ones of the form \(x_k(a)\).