[nlfzmcfz] Fermat numbers in recursive prime predecessor notation

Here are some Fermat numbers written in compressed recursive prime predecessor factorization notation (RPPFN). In a nutshell, value[(X Y)] is 1+2*value[X]*value[Y]. The notable feature is the long strings of 2s. We continue to use this Pari/GP code.

addprimes([167988556341760475137, 3560841906445833920513, 4659775785220018543264560743076778192897, 7455602825647884208337395736200454918783366342657])

f(2^2^0+1) = (

f(2^2^1+1) = (2

f(2^2^2+1) = (222

f(2^2^3+1) = (2222222

f(2^2^4+1) = (222222222222222

f(2^2^5+1) = (222222(2)) (222222() (22() (() ((2)) ((2

f(2^2^6+1) = (2222222() () (()) (222)) (2222222(2) ((((2)))) (2() (() (2))) (() () ((2) (222() ((2(() ((

f(2^2^7+1) = (22222222((()) (22222(())) (() () () (2() () (2)) (2() () (22222(() (22(222)))))))) (22222222() () () () () (2) (222((2)) ((2) (()))) (() (() (() (2() (2)))) (22(2(()) (() (((2)))) (() () () () ((2

f(2^2^8+1) = (2222222222(2() (2())) (22222(2) ((((22((2)))))) (() (() () (2) (() (2)))))) (2222222222() (2) (()) (2()) ((2() (2(2() ()))) ((((2)) (2() (() ())))) (2222() (2() (22(2) (() ()))))) (222() (2(2) ((2()) (222))) (2(()) (() (2() ())) (() (() ((((2))))))) (() () (() (2(())) (2222())) (((2()) (222(22() () (()) (() () ((2)))))))) (((() ()) (() ()) (2(()))) (2() () (222(2) (()) ((((2))) (2(())) (2(())) (() ((((2

f(2^2^9+1) = (222222222222222(2() ())) (2222222222(() ()) ((((2)))) (2() () () () () (2) ((2)) (() (2222222))) (22() () (2) (2(())) (2(2) (2)) (22() ((()) ((((2)))) (() (() ((22(()) (222)))))))) (2() (()) (2() (() () () ((2(2) (2) (2) ((2))))) (() (2) (() (()) (2(() ((2))))) (() () () (() () () (22() ()))))))) (2222222222(22() ((((2))))) (2(() (2() () (() (2))))) (22((2) (())) (() () () ()) (() (22() ()))) (2222(2) ((2)) (22222(() ())) (() (2) ((((2)))) (2(2) (()) (22(2))))) (() () () (2()) (() () (2) (222)) (22(22((2)) (2()) (() ()))) (() (2) (2) (2) (222) (22(222))) (() (() ()) (2() ()) ((()) ((() ()) ((2(2222() (()))))))) (22() () () () () (()) (2()) (22(22() (2) (()) (() (2))) (2(22() () () (2(2) (2)) (222() () ()))))) (() ((2(()))) (() () () ()) (2() () (() (2)) (() ((2) (() ())) (222(()) (()) (()) (() () () () ()) (2() () () (2) (2((

f(2^2^10+1) has some large prime factors of P252-1: 760347109 211898520832851652018708913943317 9409853205696664168149671432955079744397, but there still remains unfactored C158 130439874405488189727484768796509903946608530841611892186895295776832416251471863574140227977573104895898783928842923844831149032913798729088601617946094119449010595906710130531906171018354491609619193912488538116080712299672322806217820753127014424577. GMP-ECM performed with autoincrement up to B1=31171002. This is gettable with the number field sieve.

f(2^2^11+1) has some large prime factors of P564-1: 32488628503 1847272285831883 92147345984208191 23918760924164258488261, but there still remains unfactored C489 623519482160427323048616906708531014498894646204835383329995243920736314175010304061955059024039668210933720566293216686158106435747517268377300946271448870990313026852545958080450956436225049299162300590534361272996943566746874395767846329022831727207930728098948421726550450378478447411488755841390289363987118255143742797814710226419829858398254759293908066524722361462368335715427435081693231928614555485721774274793474319611110527957077528054659380156417680433252742848949432697352159. GMP-ECM performed with autoincrement up to B1=9300004.

These results reported by Richard P. Brent "Factorization of the tenth and eleventh Fermat numbers", found in 1995.