# # Numbers of shape: 41^(2^6) + b^(2^6) with 1 <= b < 41 # Filename: http://www.AlfredReich.com/GF.41.6.txt # Editor: Alfred Reich (zehnp@gmx.de) # Date: January 6, 2011 (last modified March 21, 2011) # # Notes: There are 40 numbers, each number has 104 digits. # Any number is completely factored. # # Numbers of shape a^(2^m) + b^(2^m) with 1 <= b < a and gcd(a,b) = 1 are called Generalized Fermat Numbers and denoted by GF(a,b,m), # compare Hans Riesel, Prime Numbers and Computer Methods for Factorizations, 1994, 2nd ed, Birkhäuser, pp102 # and Wilfrid Keller, http://www.rrz.uni-hamburg.de/RRZ/W.Keller/GFNfacs.html. # # Source: http://wwwmaths.anu.edu.au/~brent/ftp/factors/factors.gz (Richard Brent) for b=1 and 93040814849, 80350880823867160961 and 326549946497470306390924033. # 1 2 120577 93040814849 80350880823867160961 326549946497470306390924033 p41 2 362453136641 p92 3 2 257 536246116432202034423706204914086271820675385857 p53 4 1409 1532005671371009 1425562461196764291473256164737 p55 5 2 2795777 12769095795747828554391846641376280519169 p57 6 257 769 41005285527310391276781313 6978474122919234949404517505549441 p39 7 2 257 32257 127310902968785092524619554694529 p64 8 9473 179583431806081 3304437640027425447652107099649 p55 9 2 271720656877297921 1022714316292431233 p68 10 p104 11 2 641 3457 164121857 148720673312208001 p72 12 257 11734758030966759014104096513 p73 13 2 43777 302977 98197017473 p82 14 257 497106509660001979250077441 5674112283526453940174132353 p47 15 2 19457 3259755649 p90 16 20255367280731866238941441 3597657426050235492013974874241 p48 17 2 p103 18 3329 22273 1146193409 92739327283969 p73 19 2 257 1153 5725811472796417 1280220238921210627969 16173552001589179083720577 p36 20 769 86260775934709121 3080195941033472763137 p62 21 2 20064769 p96 22 641 14081 3884929 12431834608513 p77 23 2 394226595743898047233 277615741256304879685594353119731073 p47 24 257 95233 4532609 979217537 659232304897 8137382075123625602826241 p44 25 2 96769 34201679873 475969285031809 3256424433003640433281 20245975674924700098291841 p26 26 15233 5520742270265473 38266760822297560961 p64 27 2 39041 6574914817 596169938561 p77 28 257 45953 p97 29 2 113537 3740265102172940695080896724916124833793 p59 30 287233 460008585670557050469044723573334636161 p60 31 2 769 1153 26984449 30706052697601 11748148559049886627073 p54 32 769 27616100609 2775937398988236592001 p69 33 2 257 167449135153793 543478340001409 117710175087426741043234721264677889 p37 34 158273956300977281 1321849143629270017 p68 35 2 508033 2472961 23196161 1160567041 4747771393 7681581569 p55 36 44417 514049 13666058797653377 8358434075265379001473 p55 37 2 p103 38 257 26497 p97 39 2 1170971393 5994931733922223761460205636185129601 p58 40 1409 2823234466970514785567233 p76