[wzwubiuu] Mersenne primes in recursive prime predecessor factorization notation

Some large Mersenne primes in compressed recursive prime predecessor factorization notation (RPPFN), computed with this Pari/GP code. In a nutshell, value[(X Y)] is 1+2*value[X]*value[Y].

f(2^521-1) = (() (2) (2) ((2)) (222) (() (2)) (22(2)) (2(2())) ((2) (2())) (2() (2())) (22(2) (2())) (2(2()) (() (2))) (() (2) (()) (2())) (() () (2) (()) (2())) (222(2) (2()) (22(2))) (22() () (2) ((2)) (2())) (222() (2) (2222222)) (() (2) (2()) (() (2) (2) (()))) (222() (2) (2) (2) ((2)) (2())) (222(2) (2) (2) (2()) (2()) (222)) ((2) (2) (2()) (2() ()) (2(() (2())))) (2(2) (2) (2()) (() ()) (2() (2) (((2))))) (2222() (2) (2()) (22(2) (2() (() (222))))) (22222(2) (2()) ((2(() (())))) (() (2() (((2)))))) (222() (2) (()) (2()) ((()) (()) (()) ((2)) (2() () ((2))))) (22() () (2) (2()) (((2)) (() (2))) (22((22((2))))) (22(2()) ((222(()))))) (() () (2) (()) (()) (2()) (() (()) (()) (2222222(() (2) (2)) (() (2() ()))))) (2222222(2) (2()) (22((2(()))) (2() (2))) (22() (()) ((2) ((2)) (2(()))) (2(()) (22() (2() () ()) (2() () () (2) (()) (2() (()) (2((
time = 1,433 ms.

addprimes([845100400152152934331135470251, 19112684214957755703306290219340140859813072336321619])
These extra factors are from the Cunningham Project.
f(2^607-1) = (() () (()) (() (2(2) (2))) (22() (2(2) (2)) (() (222)) (22222(()))) (() (2(2) (2)) (2() () ((2()) ((((2)))))) (2() (() (((2)))) (((22(2)))))) (222() () (2(2) (2)) (22(2) (()) (() ((((2)))))) (2(() () () (())) (22() (2) (() (() () () (2)))))) ((2) (2) (2) ((2)) (() (2)) (22(2)) (2(2) (2)) ((2) (2) (2)) (22() (2) (2)) (22() () (2) (2)) (() () () () (2) (2)) (2() () () () (2) (2)) (2() (2) (2) (2) ((22((2)))))) (() () (2(2) (2)) (222(())) (222222(((22(2))))) ((222) (() (())) (2() (2) (2(2(2())) (222(2() ()))))) ((()) ((2(2()))) (22(222)) (2() () (2)) (() () (22() ()) (() () (() (2)) (22(222)) (2() (2) ((2))))) ((((2))) (2() (2)) (() (2) (() (2) (() (2() ())))) (() (222(())) (((2)) (() ())))))) (22() (2(2) (2)) ((222() (()) ((2))) (222(() () () ()) (() (2) (2) (())))) (222((2)) (((2))) (22() (2) (2(2() ())) (() (() (2)) (222(2) (2))) ((2(2) (222) (22222() ()))))) (22() (2) (2) (()) ((2)) (2(2() ())) (222() (())) (2() () (2) (222() (2)) (22(2) (())) (2() (()) (222)) (22(22222()) (22(2((
time = 31,126 ms.

addprimes([ 84462210560148142953097, 8970948423964301024591994817, 30745017857595820376423803727, 203525545766301306933226271929, 1329628131546931497103420134367, 28435302301212461494420074814087, 581211581673454706767349073071710126567])
Previous work.
f(2^1279-1) = (() () () (()) (() ()) (22() ()) (() () ((2) (()))) (22() () ((2) (()))) (() (22(2)) ((2) (()))) (22() () (2()) ((2) (()))) (2222() () (2) ((2) (()))) (((2) (())) (22() (() ((2))))) (22(2) ((2) (())) (2(222() (22((2)))))) (() () () ((2) (())) (22() (2() () (222)))) (22() (222) ((2) (())) (2((2)) (((22(2)))))) (222222(2) ((2) (())) (22(((2)) ((2)) (() () ((2))) (2(() () ((2))))))) (() () ((2) (())) ((222) (2() () ()) (222() () (((2)))) (() () (2()) (() ())))) (() (() (())) ((2) (())) (2(22() (222))) ((()) ((2()) (((2))))) ((2()) ((2(()))) ((22(2))) (22((2))))) (22() () () ((2) (())) (2222() ((((2))))) (() () ((2(22() ())))) (2() () (22() (2) ((2)) ((2)) ((2))))) (22() () ((2) (())) (() (222)) ((22(2(())))) (2() (()) (222)) ((2(2() (2(222) (2(()))))) (() (2() () (2) (2) (2) ((((2)))))))) (22222() () ((2) (())) (() (() (2() ())) (() () (222) (2((2)) ((() ()) ((22((2)))))))) (() (2() (2) (2() (2)) ((2) (())) (2() (2) (() ()) (() ()) (2(2())))))) (22() () (() ()) ((2) (())) (2(()) (() () () (()))) (22() () ((2) (22222()))) ((() ((2)) (222)) (2(222) (22(2) (2())) (2() (2(2) (2))) (22() (2() () (222)))))) (() () ((2) (())) (22222(2) (2) (()) (((22(2)))) ((2() ()) ((2(2()))))) (222(()) ((2)) (2222((2)) (22222()) (() () (2) (2() ())) (() () (2(() ((2)) (2(2())))))))) (() () (2()) (() ()) ((2) (())) (2() (222) (2() () () (22(2) (() ())))) (222() ((2)) (222222(2)) (2() () (((2))))) (() (2) (2()) (((2))) ((2) (2) (2) (() (() () (() (2()))))))) (() () (()) ((2) (())) ((2) ((2)) (22(()) (222))) (2(() ()) ((((2)))) (2(()) (() (222() () (2()))))) (2() (() (2) (2)) (((2) (2()))) (() () (()) (() (2)))) (2() (2) (22(()) ((2) (((2))) ((2) (2() (() (2)) (() (222)))))))) (() () (2) (() ()) ((2) (())) ((2(2()))) (() (() (222) (2(())) (() (2)) (() (()) (2())) (2(()) (2() (222) (() ()))))) ((()) ((2) (2) (2(()))) (() () (2()) ((()) (((2))) (22() ()) (2222() () (()) (((2))) ((((2)))))) (() ((2)) ((2() () ()) ((() (2)) ((222(() ((2)) (() ((2))))))))))) ((() (22() (2) (2))) (() ((2) (())) ((() ()) ((22((2)))))) ((2) (()) (() (((2)))) (22() ((2()) (2()) ((((2)))))) (() ((2)) (() ()) (22(()) (() (())) (2() (()) (() ())) (22(2) (2() () (() (2) (((2
time = 51,387 ms.

addprimes ([20837062885084633147, 258977744356523549983, 301311116540899114446723859201, 883533090360873723903538281367, 60716055995924620572719451834239, 115092709908691724294367540635071, 460233616861852066165180033789571, 1636198597169607245088331633873083979, 1165410983472064229956077783033756619354709, 19755740081951910036006278827509875120092863638283602681, 79073321945266228838262115990524608069915168947411523193620486700648001, 13307293644989926855503183079276784290565688582692135767130945147226822516757076633431114901443896343698334635540215594678754538286654095364149897])
Previous work.
f(2^2203-1) = (() () (()) (() (() (2() (2)))) ((222) (() (2() (2)))) (() () () (() (2() (2)))) (() (2()) (() (2() (2)))) (22() () () (() (2() (2)))) (22222() () (2) (()) (()) (() (2() (2)))) (22() (() ()) (() (2() (2))) (2(2) ((2()) (((2))))) (() () (2) (222() () ()))) (22(2()) (22() ()) (2222((2))) (() (2() (2))) (() (2) (2() (()) ((2(())))))) (() (() (2() (2))) (2() (2) ((2) (()))) (2(2()) (2() ()) (() () () (2()) (() ()) (() (((2) (()) (() (222)))))))) (() (2() (2)) (22() ()) (() (2() (2))) (() (() ((2))) ((() () (())) (22(2(2) (2)))) (222() (2() () (2) (2() ()))))) (2222222() () () (2) (2) (() (2() (2))) ((()) ((22(2)))) (2(2() (22(222)) (2(22() ())))) (22() (()) ((22(2))) (() (2) (2)) (() () () (())) (2222((22(2)))))) (() (2() ()) (() (2() (2))) (22(2() (2)) ((222) (2() ())) (() (()) (2() () (2)))) ((() () (2) ((2))) (((((2))) (2() () ((2(2())))))) (22() () (2(()) ((2)) (2(2() ())))))) ((2) (()) (() ()) (() (2() (2))) (() (22() () (2()))) (222() () (()) (() () () () () ((2))) (22(() ()) (2(2) (2) (() (2) (()))) (() (2) (((2(()) (2()) (2()))) (2() (() () (() (2()))))))))) (() (2() ()) (() (2() (2))) (2((2)) (() (222) ((2(()))) (() ((2) (2()) (() (()))))) (2(() ()) (2(2())) (222(())) (2() ((2) ((2)) (222)))) (22() (()) (222) ((22(2))) (2(2222())) (((2)) (2() () ()))))) (22(2) (222) (() (2() (2))) (() ((2)) (2()) (2() () (222() () () (2) ((2)))) (() () (2) (2(())) (() ((2(2())))))) ((()) ((((2)))) (22() (2) (2(()) (() (()) (((2))) (22(2))) (() (2) (() () (())) (((22((2)))) (() ((22(2))))) (22(2) (2) ((22(2(2) (2))))) ((2(2) ((2) (2() ((2(())))) (() (() ()) (() (222))))))))))) (22222() (2) (2) (2) (() (2() (2))) (() ((22(2(()))))) (2(2) (()) ((2(() () () ())))) (() (2) (() (() (())) (() ((2))) (22(((2)) (2()) (2()))) ((222(((2))) (() (2)) (2((2)) (222(()))) (2() () (()) (()) ((2(()))) (22() (2) (2))))))) (2() (222) ((2(2()))) (2() (2(()))) (22() () (((2)))) (2(2() () (22() (22((2)))) (() (()) (22((2)))))) (() () () (() (2()) (() (2() ())) (2(2(())) (2(2) (2))))))) (22() () () (() (2() (2))) (2() ((2(())))) (() () (() (((2))))) (() (()) (() (2)) (2(2())) ((2(2()))) (() (2() ())) (2(2) (2) (2) (2(()) (()) (2() ())) (22() () ((()) ((((2))))) (() () ((((2) (2222() (2(())) (2222() (()))))))))) (2(()) (() (((2)))) (() () (2(2())) (2((((2)))) ((2(2()))) (2() () () (2)) (2() (2) (2) (2(2()))))) (() () () () (2) (2() (2(2) (2))) (2() () () ((2) (222) ((2) (2() (2)) (2() () ())))) (2() (2() ((22(2)))) (2() () (() () () (2) (2())))))) (2(()) (()) (()) (2() () ((2))) (() () (2222(2) ((2)) (2(2()))) ((((2))) (() (())) (2() (22(2(())))))) (2() (22() (2) (2) (())) (2(2) (()) (()) ((2))) (2() (2) (()) (2() (((2))) (() () (2(())))))) (22() ((2(() (2)) (() (())))) (2(2222() (()) (2()))) (22() () ((2)) (2() (2))) (222(2() (2() (2) (((2)))))) (() (() (22() ())) (() (() ()) (2() ()))))))) (() (() ((2))) (() () ((2))) (() (2() (2))) (22(()) ((2))) ((222) (2() ())) (2222() (2(())) (2(2())) ((2()) (222)) (((2)) (() (()))) (() () () ((2)) ((2(2())))) (2(22() ()) (() (2) (()) ((2)))) (((2)) ((((2)))) (2() (()) (() ()) ((((2))))) (22() (2()) (2() ((2(()))))) (2() ((2)) (2(2())) (22() (2) (222(2() ())))) ((222) (22() (() ()) (2() ()) (() (2) (()) ((2(()))))))) (() (2() () (222)) (() (()) (2() ()) ((((2))))) (2() (2(2() ()) (2() (() ()))) (((((2)))) (() () (()) (())))) (() (2) (222() (2) ((22(2))) (() () (222() (()))) (22(()) ((2()) (2(() (2())))))) (22222(22() () () ((2)) ((2) (())) (22() (2())) (() ((2) ((2)) ((2)) (() ((22(2(())))))))))) (2222(()) (()) ((2)) (2(2) (2)) (() () (() () (2() (((2))) (22(2) (()))))) (() () () () (()) (2()) (2()) (() ()) (222(())) (2(() (())))) ((22() (222)) (2(2) ((2) (())) (22() () (22(2))) (() (2() () ((22222(2) (2)))) ((()) (()) (()) ((2(2) (2)) (2() () (2
time = 1min, 58,139 ms.

addprimes([276696631250953741, 2416923620660807201, 3011347479614249131, 1491477035689218775711, 23480412082098913326841, 25349242986637720573561, 29034057164920993379000074993, 64326196787727903551977150861, 3435950210316335724157758000789490561, 15653990705896313547269237220041169361, 153787279330237476887106331233239525756635010497681, 3297516038750209158351848769787405523717142645335063417537342353153])
Previous work part 1, and part 2 (including general number field sieve).
f(2^2281-1) = (() () (2) (2) (()) ((2)) (2()) (222) (() (2)) (22(2)) (2() (2)) (() (2) (2)) ((2) (() ())) (2() (() ())) (222() (2)) (() (2) ((2))) (22() (() ())) (() (2) (() ())) (22(2) (() ())) (22222(() ())) (22() (2) ((2))) (22() (2) (() ())) (222() (2) (() ())) (22() (() ()) ((2) (()))) (22222() () (2) (() ())) (222() (2) (2222222)) (222() () () ((2)) (() ())) (2() (2) (2) (() ()) (((2)))) (2222(2) (()) (()) (() ())) (22() (() ()) (2222((2)))) (() () (()) (() ()) (22() ())) (2() (2) (() ()) (() () () ())) (2() (2) (()) (() ()) (() (2))) (() () () (()) (() ()) (22() ())) (222222222() () () (() ())) (() (2) (() ()) (2() (()) (((2))))) (() (() ()) (() () () (2(()) (())))) (222() (() ()) (2(() ((2) (2() (2() (2))))))) ((2) (2) (2) (2) (()) (() ()) ((2) ((2)) (((2))))) (22222() () () (2) ((2)) (() ()) (2(()) (() (2())))) (222() (2) (222) (() ()) (2() (() ())) (2222222)) (222() (2()) (() ()) (() ((2) (() (2(())) (22() ()))))) (() () (2) (() ()) (22(22222((2)) ((((2)))) (() ((2)))))) (222222222() () () () (() ()) (() () (()) (() ()) (22() ()))) (222() (222) (() ()) (2(22222() ())) (22((((2))) (2() (2) (() (2)))))) (22() (2) (() ()) (2() () (2()) (() ()) (2(()))) (2() (() (2() (2))) ((22(2) (()))))) (2() () (2) (() ()) (2222(2) (() ())) (22() () (()) (() () ((2)))) (2() (2) (2(2) (2()) (222)))) (2222(2) (2) (() ()) ((2) (((2))) (() (2)) (2() (22((2) (())))) (22(222(() (2)) (() (22(222))))))) (() (2) (() ()) (2(2()) (222222() ())) (2() () (((2) (()) (()))) (() (() () () (2) (2(())) (2(2())))))) (() (2) (() ()) (() (2() (2))) (22222() ()) (((222(()))) (2() (222(2) (2)))) ((2(()) (2() () () (2) ((2)) (222))))) (22() (2) (()) (2()) (() ()) (22(2) (2222())) (() () (2()) (() ())) ((() ()) (() ()) ((2) (()))) (22((2(()))) (() () () (2)))) (22() (2) (() ()) ((((2)))) ((()) (2()) (((2))) (() () (()))) ((()) ((() ()) (2(22222() ()))) (22() () () () (2()) (() (()))))) (222() (() ()) (() ()) (2() ((2(())))) (2() (()) (() (() ()) (2(())))) (2(()) (2(2() ())) (2(2) (()) (((2))))) (222() () (() () () (2)) (((2) (() () () () (2)))))) (2() (2) (() ()) (2(() (2()))) ((2222(22(2) (2) ((2(())))))) (22() () () (2) (() () (2()) (222() () (()))) (() () () (2()) (222) (((() ()) ((2(() () (()))))))))) (222222() (2) (() ()) (((2))) (2(() (()))) (22() () (()) (()) (() () () (2)) ((2) (2()) (() (()))) ((((2) (2()) (2() (2()))))) (22(() ()) ((() (2)) (22(2)))) ((2) (2() (() ()) (() ()) (22(2)) (() (2())))))) (222() (2) (() ()) (222222() (2) (() (22() () (2) (2) (2((2)) (((2)))))) (2(2) (() (2)) ((222(()))) (() () (2(())))) (222(() (222)) (2() () () (2() (2())) (() (2) (()) (22() () (2) (() (2) (2) (())))))))) (222() (2) (() ()) (() () (())) ((2) (() ())) (2() () (((2)))) (2222(2) (2) (()) (((2))) ((((2)) (() ()))) ((2) ((2)) (2()) (2() ()) (2222(2(2() ())) (() () ((2))) (22() (() (222) ((22(2)))))) (2() (2) ((2(() (2()))) (() (() (2) ((2))))) (() (() (((2)))) (2() (() (2) (() ((2)))))))))) (2222() (2) (() ()) (((2))) (22(2)) (22() (222) (() () (())) ((2() () (222)) ((2) (()) (2())))) ((() () (2) (2())) ((() (2) (2) (2())) (2() () (22222(() (())))) ((()) (2()) (222) (2222())))) (2222222() (() ()) (2(2) (2)) (() (() (2())) (() (2()) (2()) (222)) (() (2) (() ((2) (() (2(() (())))))))) (2(22(2)) ((()) ((2)) (2())) (22222() (222) ((()) (() (2) (2)) (2222((2)))))) (2(((2))) ((2(2()))) (() (22((22(()) (2(2())))))) (2() (2(())) (2(2())) (((2) (2()))) (2() (((()) (222)))) ((((2))) (2() ()) (2() (2(())) ((((2)))) (() (2(() ((2)) (2(2())))))))))) (22(()) (() ((2)) (222) (222) (((2))) (2(() (2)) (222() () ())) (() (2) (2) (() ((2))) (2() (() () (()) (()) (2(()))))) (2(() ((2(()))) (2() (2(2() ()))) (() ((2) (() (2))))))) ((((2))) (() (2() (() (2)))) (2(2() (2)) (() ((2)))) (2(2) (22(2)) (222() (2))) (22(2) (222) ((() ()) (2222222))) ((2) (() (2() ()) ((2(2()))) (() (2) (()) (())) (() (()) (()) (2(() ((2) (2() () (
time = 2min, 49,586 ms.

addprimes([253262965868307257488778209946038131361, 5722137022002067824248227975095857749151312827809388406962346253182128916964593])
The following number is not prime, but has had previous work.
f(2^757-1) = (() (2() () () (())) (222() () () (2))) (() () (2) (2() () () (())) ((2()) (2(() (2()))))) (222(2()) (2() () () (())) (22((((2)))) (() (2) (2() () ()))) ((222) ((22(2))) (() () ((2))) (2(2) (()) (2(() (2())))) (() ((2) (())) (22() () ((2) (()))))) (2222(() (())) ((2(()))) (22() ((2)) ((((2)))) (2() (222) (((2))) (2(2222())) (() (2) (2) (2))))) (222(22(2)) (2(2) (222222(2))) (2(2()) (() () () (()))) ((()) ((() (2)) ((2) (2())))) (() (()) (2(())) (22() (2) (((((2) (2())) (2(2) (2(2()))) (2(() ()) (2() () ((((2))))))))))))) ((2() () () (())) (() () (2) (2222((2)))) ((222(22() ((2)) (((2))) ((()) ((22(2))))))) (() () () () (2()) (2(22() ())) ((2) (2) (2(222222() () ())))) (2(() () (((2(() () (()))))) (2222() () () () (2) (2) (2) (2() (2))))) (2222() (2) (() (())) (22() ()) (22() (2) (() (() ()) (2(2())))) ((2) (222() (2() ()) (2() ()) (2(2222())))) ((2() ()) (2(2()) (() (2() () ((2)))) (2() ((2)) (() (2() (((2)))) (22(2) ((2(()))) ((2) (() ())))))))) (2((2) (())) ((2() ((2) (()))) (2(() () (22(2))))) (2() (2) (2()) (2(2) (2(2()))) (22(2() (22(2)) (22(2)) ((()) (222))) (2() (2() (2) (()) (() ((2)) (2(2()))))) ((() (())) ((2((2) (())) ((2(2()))))) (() (2()) (2()) (((2))) ((222222((2))))) (2() ((((2)))) ((2) (2())) (() () () ()) (22222() ((2
time = 7,965 ms.