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Linux for a Turion X2
(sh.itjust.works)
From Wikipedia, the free encyclopedia
Linux is a family of open source Unix-like operating systems based on the Linux kernel, an operating system kernel first released on September 17, 1991 by Linus Torvalds. Linux is typically packaged in a Linux distribution (or distro for short).
Distributions include the Linux kernel and supporting system software and libraries, many of which are provided by the GNU Project. Many Linux distributions use the word "Linux" in their name, but the Free Software Foundation uses the name GNU/Linux to emphasize the importance of GNU software, causing some controversy.
Community icon by Alpár-Etele Méder, licensed under CC BY 3.0
Like some of the other comments, if you really need a DE then maybe give XFCE or LXQt a try. The distro itself won't matter too much in your scenario.
I do have an old laptop that has run Debian/Ubuntu + Gnome fine, not at all fast but usable for my needs. Mines has 4 GB RAM, get the feeling that going under 4 GB may be a bit much.
Otherwise Linux is perfectly usable without a DE if you're willing to stick to the terminal for all your usage.