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You can simply paste LaTeX without logging in or something else. Just go to the new paste page and enter your LaTeX. If you mark your paste as private it gets an 40 bit long unique hash which is hard to guess and the paste won't appear in the recent changes list. But if you give your paste a tag it will appear in the tag cloud/list. Nevertheless you shouldn't paste private information in this pastebin because someone might get the link to a private paste by sniffing your internet connection etc.
TeXbin automatically saves your name in a browser session together. That data is deleted as soon as you close your browser or the session expires. Currently there is no way to save your user information for longer time periods or to register a nickname just for you.
You can give your paste a whitspace delimited list of tags. An automatic completion helps you finding tags that are already in use. Those tags can help you to find your snippet again if you think it's useful. Please don't tag your LaTeX with tags that have nothing to do with your sourcecode.
If you click on the "reply to this paste" link on a detail page you can change the paste and save a new version. TeXbin saves version information then so that you can compare the paste with the previous version.
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