It is an important, and potentially deeply infuriating, stage in any computational number theory problem to type in the answer you've obtained to Google and discover that it was originally obtained by Noam Elkies twenty years earlier.

Ah yes. Much like looking up your shiny new algorithm in the index of Knuth.

Are there really {m}any new algorithms to be found?

Incidentally, your quartic surfaces page seems rather unclearly written in places. It took me a while to twig that the quartic surface equation given in the page title was not in fact a typo but was talking about rational rather than integer solutions. Then you shift from that representation straight to adding a t⁴ term and talking about integers, and suddenly shift back again in the hyperplanes section.

With a bit of thought it's not impossible to work out what you're talking about, but I think I'd prefer that you only require thought of your readers in the bits that are actually supposed to require thought :-)