How do I submit a binary?

You can not submit a binary file, just source packages.
See https://help.launchpad.net/Packaging

Oops i think my comment is unrelated sorry lol still a bit tired, please delete it, i cant.

You have to understand the versioning policy of Ubuntu:
Most of the packages that are provided in Ubuntu have been taken over from Debian. At a certain pint in the development of a new Ubuntu release, the packages are taken over from Debian in the version that they have there. If there are updates to a package later, that are made available in Debian after the iport date to Ubuntu, theser verseion will only be made available with the next Release of Ubuntu.
https://launchpad.net/ubuntu/+source/gmp-ecm shows, that the gmp-ecm version avialable with Natty, Oneiric and Precise is 6.3, and for Quantal the current development is working on 6.4.2. Normally there is no plan to provide newer version (like 6.4.2) on older releases (like Natty).

There some exceptions:
For a specific list of packages, there are updates to current version provided also for older releases (e.g. firefox).
For some software packages sometimes 'backports' are provided (e.g. a newer version of inkscape provided also for Natty).

If you need a certain newer version of a package you can:
1. upgrade your Ubuntu version to the release that provides the right package version (in sereval cases not possible at all, or undesired)
2. search if there is a PPA (personal package archive) on Launchpad for your Ubuntu release that provides that version.
3. download the source of the package and build the package yourself (I understand that you have done this already)
4. manually install the newer package version for the higher Ubuntu release on your lower Ubuntu release (this can work well, but can also miserably fail and render your system completely unusable - not recommended, do at your own risk).

I understand that you have built the package with current version against the Natty release. If you want to make that package available for other Ubuntu users, you can provide that package in a PPA. You have to note that also for a PPA no binary files are provided to Ubuntu, but only adapted source files, that are re-compiled on Ubuntu servers, and the resulting binary files are then made available for download.

If you want to provide your version to others via a PPA read https://help.launchpad.net/Packaging and https://help.launchpad.net/Packaging/PPA to see how to start.

"Normally there is no plan to provide newer version (like 6.4.2) on older releases (like Natty).

There some exceptions:
For a specific list of packages, there are updates to current version provided also for older releases (e.g. firefox).
For some software packages sometimes 'backports' are provided (e.g. a newer version of inkscape provided also for Natty)."

That seems kinda stupid to me to only update some packages. What makes those packages special compared to others? While a release is supported, it makes sense to me that all packages should be updated to their most recent versions.

As for making a PPA, that seems like a lot of trouble to go to, especially since no one will realize there is a PPA with the newer version available.

As for why I got this to compile on my own when originally I wanted to use the package: From its name you can see it requires GMP to compile, which meant I also needed to compile that; I wound up having to do that for another program anyways, so I just compiled GMP-ECM along the way.

Thanks for the reply in any case, it certainly answers my question (I don't like the answer, but that's certainly not your fault).