There are thus natural numbers whose binary representations we can easily inscribe, but for which no human mathematician or foreseeable computing device can carry out the trial division algorithm. This again might not seem particularly troubling as this algorithm is indeed ‘naive’ in the sense that it admits to several obvious improvements – e.g. we need only test \(x\) for divisibility by the numbers \(2, \ldots, \sqrt{x}\) to find an initial factor, and of these we need only test those which are themselves prime (finitely many of which can be stored in a lookup table). Nonetheless, mathematicians have been attempting to find more efficient methods of factorization for several hundred years. The most efficient factorization algorithm yet developed is similar to the trial division algorithm in that it requires a number of primitive steps which grows roughly in proportion to \(x\) (i.e. the size of its input, as opposed to the length of its binary   representation).[6]     A consequence of these observations is that there exist concretely inscribable numbers – say on the order of 400 decimal digits – with the following properties: (i) we are currently unaware of their factorizations; and (ii) it is highly unlikely we could currently find them even if we had access to whatever combination of currently available computing equipment and algorithms we wish.