## Monday, October 01, 2018

### [uhpeqtdl] Small Fermat number factors

We list the smallest factors of some Fermat numbers.  These factors constitute compact proofs to the compositeness of each Fermat number.  Additional larger factors are not useful for proving compositeness.

We choose a novel encoding to emphasize compactness.  For each entry (n,x,y): 2^2^n+1 is divisible by 1 + (3+2*y) * 2^(n+2+x)

Edouard Lucas showed that x>=0.  We conjecture that y>=0 (this seems easy to prove).  y=0 at n=38 (and others larger).

(0,Prime)
(1,Prime)
(2,Prime)
(3,Prime)
(4,Prime)
(5,0,1)
(6,0,534)
(7,0,58251551882320)
(8,1,302472256237)
(9,5,17)
(10,0,5564)
(11,0,18)
(12,0,2)
(13,1,20682941)
(14,0,892090498909563978798187208821078272555440547357)
(15,4,288)
(16,1,786)
(17,0,29625927)
(18,0,5)
(19,0,16813)
(20,Composite with no known factor)
(21,0,267343)
(22,0,1926979601222033828766816104)
(23,0,1)
(24,Composite with no known factor)
(25,2,24205)
(26,1,71581)
(27,1,70506)
(28,6,12854659685)
(29,0,560023)
(30,0,74519)
(31,0,2731780735650)
(32,0,738)
(33,Character unknown)
(34,Character unknown)
(35,Character unknown)
(36,1,1)
(37,0,637719231)
(38,1,0)
(39,0,9)

Some Pari/GP code for verifying factors:
val(n,x,y)=1+(2^(n+2+x))*(2*y+3)
test(n,x,y)=lift(Mod(2,val(n,x,y))^2^n+1)

For a much more comprehensive list, see http://www.prothsearch.com/fermat.html.