Quelltext der Seite PrimeGrid/Generalized Fermat Prime Search

'''Generalized Fermat Prime Search''' sucht nach Primzahlen der Form '''GF(''n'', ''b'')''' = '''''b''<sup>''N''</sup>+1''' mit ''b'' > 1 und ''N'' = 2<sup>''n''</sup>. Im Spezialfall ''b'' = 2 spricht man von Fermat-Primzahlen. Man kann leicht zeigen, dass ''b'' eine gerade Zahl und ''N'' eine Zweierpotenz sein muss, damit ''b''<sup>''N''</sup>+1 prim sein kann:

Ist ''b'' ungerade, so gilt ''b'' &equiv; 1 (mod 2). Dann ist aber

''b''<sup>''N''</sup>+1 &equiv; ((1&nbsp;(mod&nbsp;2))<sup>''N''</sup>&nbsp;+&nbsp;1)&nbsp;(mod&nbsp;2)&nbsp;= (1&nbsp;+&nbsp;1)&nbsp;(mod&nbsp;2)&nbsp;= 0&nbsp;(mod&nbsp;2)

d.h. ''b''<sup>''N''</sup>+1 ist durch 2 teilbar und niemals prim.

Ist ''N'' keine Zweierpotenz, so lässt sich ''N'' = ''r''&sdot;''s'' mit ungeradem ''s'' schreiben. Es ist dann

''b''<sup>''N''</sup>+1 = (''b''<sup>''r''</sup>)<sup>''s''</sup>+1<br>
&equiv; ((-1&nbsp;(mod&nbsp;''b''<sup>''r''</sup>+1))<sup>''s''</sup>&nbsp;+&nbsp;1)&nbsp;(mod&nbsp;''b''<sup>''r''</sup>+1)<br>
= (-1&nbsp;+&nbsp;1)&nbsp;(mod&nbsp;''b''<sup>''r''</sup>+1)<br>
= 0 (mod&nbsp;''b''<sup>''r''</sup>+1)

d.h. ''b''<sup>''N''</sup>+1 ist durch ''b''<sup>''r''</sup>+1 teilbar.

{{PGSubprojekt
|Status=<span style="color:#00ff00;">'''aktiv'''</span>
|Homepage=http://pagesperso-orange.fr/yves.gallot/primes/index.html
|Beginn=1999
|PGBeginn=5. November 2009
|PGForum=http://www.primegrid.com/forum_thread.php?id=1528
|Suchbereich=''N'' = 2, 4, 8, 16 mit ''b'' < 1G<br>
''N'' = 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216 mit ''b'' < 100M
|App=
|AppBadge=
|Sieve='''GFNSvCUDA''' ([[manuelles_Sieving|manuell]])
|SieveBadge=
|Primtest='''Genefer''' (via BOINC)
|PrimtestBadge=genefer
|SGForum=<vbthread id=3736>Generalized Fermat Prime Search</vbthread>}}
==Funde==
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