As of 2007, only the first 12 Fermat numbers have been completely factored see: http://www.prothsearch.net/fermat.html.
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As of 2007, only the first 12 Fermat numbers have been completely factored see: http://www.prothsearch.net/fermat.html. Fermat conjectured that all were prime; however, it is now believed that all but the first four are composite.