a>b integers, gcd(a,b)=1 or a and b are the roots of the quadratic equation x2-Sx+P=0 (imaginary?) extended Generalized Mersenne numbers xGM(a,b,n)=(a^n-b^n)/(a-b) extended Generalized Fermat numbers xGF(a,b,m) = a^2^m + b^2^m
Phi2(a,b) = (a2-b2)/(a-b) = a+b = L1(a,b) Phi3(a,b) = (a^3-b^3)/(a-b) = a2+ab+b2 = L2(a,b)+ab Phi5(a,b) = (a^5-b^5)/(a-b) = a4+a3b+a2b2+ab3+a4 = L4(a,b)+abL2(a,b)+a2b2
F(m+1)-2 = F(0)F(1)F(2)F(3)...F(m) Fermat numbers are coprime ----------------------------- a^2-b^2 = (a-b)(a^1+b^1) a^4-b^4 = (a^2-b^2)(a^2+b^2) = (a-b)(a^1+b^1)(a^2+b^2) a^8-b^8 = (a^4-b^4)(a^4+b^4) = (a-b)(a^1+b^1)(a^2+b^2)(a^4+b^4) .... a^2^(m+1)-b^2^(m+1) = (a-b)(a^1+b^1)(a^2+b^2)(a^4+b^4)...(a^2^m+b^2^m) xGF(m+1)-2b^2 = (a-b)xGF(0)xGF(1)xGF(2)...xGF(m) If b=1 and a is even then xGF is odd, hence coprime If b=1 and a is odd then xGF is even, xGF/2 is odd and coprime (2,1) Fermat (3,1)/2 (3,2) (4,3) (5,1)/2 (5,2) (5,3)/2 (5,4) (6,1) (6,5) (7,1)/2 (7,2) (7,3) (7,4) (7,5) (7,6)
a=3 b=1 (GF3) always (except m=0) is divisible by a single 2 (why not more?) m 3^2^m+1 0 2 * 2 1 2 * (5) 2 2 * 41 3 2 * (17) * 193 4 2 * 21523361 5 2 * 926510094425921 6 2 * 1716841910146256242328924544641 7 2 * (257) * 275201 * 138424618868737 * 3913786281514524929 * 153849834853910661121 8 2 * 12289 * 8972801 * 891206124520373602817 * P(90) 9 2 * 134382593 * 22320686081 * 12079910333441 * 100512627347897906177 * 185203545384014444998415700339182963094565346115682981302539679248192554448695865898272140289 * P(101) pattern of (5)(17)(257) stops at (65537) after that, p=F(m) is not prime and Fermats little theorem 2^(p-1)===1(mod p) does not work This is actually the Pepin test!! this pattern was mentioned by Fermat in one of his letters! 15 2 * (65537) * 786433 * 3041503300933451777 * 951901181416549122049 * C(15584) 31 4638564679681 * 273963078909953 * C()
a=5 b=1 (GF5) always is divisible by a single 2 m 5^2^m+1 0 2 * 3 1 2 * 13 2 2 * 313 3 2 * (17) * 11489 4 2 * 2593 * 29423041 5 2 * 641 * 75068993 * 241931001601 6 2 * 769 * 3666499598977 * 96132956782643741951225664001 7 2 * (257) * 23653200983830003298459393 * 24171717725330873572798545219226642215966994254472458802413313 8 2 * 1655809 * 101199664791578113 * 4563566430220614493697 * 12025702000065183805751513732616276516181800961 * P(88) 9 2 * 19457 * 23606273 * 236554494714698753 * C(329)
a=3 b=2 m 3^2^m+2^2^m 0 5 1 13 2 97 3 (17) * 401 4 3041 * 14177 5 1153 * 1607133116929 6 769 * 121899667073 * 36629538145348481 7 (257) * 45876204582640401445607833244277975113391731388650867226881 8 72222721 * 343200070657 * 2226198380033 * 3376663028737 * 1839605176202823817996787333633 * 405549420455750246193993361998354279613273199617 9 4043777 * 57987375533057 * 406297848379393 * 296909426637032277938768757025793 * 1440276254252769698681054259953238091664449537 * 2236335363090199099071989377913382509011409921 * 212085278921429005793934248630969041682071162298569605814071345111198398726196355614721 10 [2330249132033] * 10176954088500686156890644481 * C(449) 11 [625483777] * 4910460769317099419184178986913042832453633 * C(926) 12 [286721] * [1810433] * C(1943) 3 2 10 [2330249132033] 1137816959*2^11+1 3 2 11 [625483777] 76353*2^13+1 3 2 12 [286721] 35*2^13+1 3 2 12 [1810433] 221*2^13+1 3 2 13 6106257270912679 14 2004 Fougeron (ECM) 100044919126633332737 3 2 13 43270447114800194142542157469491 15 2023 Doescher (ECM) 1417886011057772761662821415960281089 3 2 14 2542706332890435 16 2004 Fougeron (P-1) 166638802232307548161 3 2 17 30344817572728262857177 18 2021 Nohara (ECM) 7954711857785277738431807489 3 2 18 18759640756820266421 19 2021 Nohara (ECM) 9835454533111783841333249 3 2 14 2059994249 15 2004 Fougeron 67501891551233 3 2 27 9170030349955 28 2021 Doescher 2461561278524010004481
a=4 b=3 m 4^2^m+3^2^m 0 7 1 5^2 2 337 3 (17) * 4241 4 4338014017 5 11969 * 1541364950613953 6 337153 * 1009281751473729386230842057136129 7 (257) * 450552876409790689547687014580821320521956599525911399679646600325891979521 8 C(155) 9 52996750337 * 59577991748609 * C(284)
a=5 b=2 m 5^2^m+2^2^m 0 7 1 29 2 641 3 (17) * 22993 4 97 * 1573071713 5 193 * 102593 * 1175885676400129 6 274568286337 * 1974376186976624976974230015921793 7 (257) * 115201 * 152833 * 79280897 * 2076762034602227299841 * 394456169989551214780703633628899655319959827713 8 9
a=5 b=3 m 5^2^m+3^2^m 0 2^3 1 2 * (17) 2 2 * 353 3 2 * 198593 4 2 * 97 * 786757409 5 2 * 143797249 * 80957968182017 6 2 * (257) * 290320001 * 3632789233881076244813676034342529 7 2 * 7681 * 36097 * 259841 * 21440257 * 26556161 * 3582108415820542080084579165968674484267701963220750596627457 8 9
a=5 b=4 m 5^2^m+4^2^m 0 3^2 1 41 2 881 3 (17) * 26833 4 4801 * 32677121 5 641 * 5953 * 59925889 * 101900353 6 1699132282369 * 319046040234900998188151306914049 7 (257) * 1719885157889 * 664856597964527775591232190673009405184495378793489594446703567585557440257 8 176129 * 2514433 * 446202881 * C() 9 C()
a=6 b=1 (GF6) m 6^2^m+1 0 7 1 37 2 1297 3 (17) * 98801 4 353 * 1697 * 4709377 5 2753 * 145601 * 19854979505843329 6 4926056449 * 447183309836853377 * 28753787197056661026689 7 (257) * 763649 * 50307329 * 3191106049 * 2339340566463317436161 * 2983028405608735541756929 * 18247770097021321924017185281 8 18433 * 69615986569139423375849495295909549956813828853888948633601 * P(137) 9 80897 * 3360769 * 12581314681802812884728041373153281 * 3513902440204553274892072241244613302018049 * P(311)
a=6 b=5 m 6^2^m+5^2^m 0 11 1 61 2 (17) * 113 3 2070241 4 2973697798081 5 193 * 577 * 645313 * 9104321 * 12199937 6 (257) * 1189782673537 * 207149065298675818568214979118531969 7 3329 * 8253953 * 3108109313 * 29759533890817 * 28232308132114188879656724422657 * 55913344106796247401204263131719169 8 9
a=7 b=1 (GF7) m 7^2^m+1 0 1 2 3 4 5 6 7 8 9
a=7 b=2 m 7^2^m+2^2^m 0 1 2 3 4 5 6 7 8 9
a=7 b=3 m 7^2^m+3^2^m 0 1 2 3 4 5 6 7 8 9
a=7 b=4 m 7^2^m+4^2^m 0 1 2 3 4 5 6 7 8 9
a=7 b=5 m 7^2^m+5^2^m 0 1 2 3 4 5 6 7 8 9
a=7 b=6 m 7^2^m+6^2^m 0 1 2 3 4 5 6 7 8 9
a=8 b=1 m 8^2^m+1 = (2^2^m)(4^2^m-2^2^m+1) 0 1 2 3 4 5 6 7 8 9