In “Pi Approximations“, line 58, Weisstein mentions one by Shanks (1982) that differs by a mere as,

where *u* is “…a product of four simple quartic units”. Frustratingly, he doesn’t give *u* but I eventually found the primary source online. Hence,

where,

with a slight modification by this author since Shanks didn’t realize the first two quartic factors were in fact squares. (The *product* of the last two factors is also a square.)

A cute thing about these numbers is that their defining polynomials are *palindromic*, the same read forward or backward. For example, the first factor (unsquared) is the root of,

*Author’s note*: Daniel Shanks (1917-96) was a mathematician best known as the first to calculate pi up to 100,000 decimal places, as well as for his book, *Solved and Unsolved Problems in Number Theory*.

In general, Shanks’ approximation belongs to the family,

where is the *Dedekind eta function* and, for *m* a positive *odd* integer, then *x* is an *algebraic integer *that is the root of an equation *P*(*x*) with palindromic (if unsigned) coefficients. For appropriate *k*, then *P*(*x*) has degree equal to the *class number* h(-2m). Furthermore, it is* *solvable in radicals*. *

For example, given prime *m*, with 2*m* = {10, 14, 26} which has class number 2, 4, 6, respectively, then *k* = 12 and,

and so on. It then is a simple matter to take the natural logarithm of both sides to bring down pi and have a relation of form,

Shanks chose 2*m* = 3502 since *d* = 4(2*m*) is the largest *fundamental discriminant* *d* divisible by 4 with class number *h*(-d) = 16. Here is a list of of *d* with small class number. You can calculate it (among many other things) in www.wolframalpha.com simply as,

*ClassNumber*[*Sqrt*[-*d*]]