3 No more known candidates thanks to François

For dealing with large potential primes a good choice is openpfgw

Using openpfgw I finished the list to $1 3466 917$ in about 20 minutes without finding new primes.

[added] The only prime perfect + 1 candidate from the known Mersenne primes is for $M_{20996011}$ - I am running ECM factoring on it.

[later] François Brunault found that $M_{20996011}$ is divisible by $1552147$ which settles the question for the known perfect numbers.

Here is the log:

./pfgw64 -f10 -lmer1log.txt /tmp/mer.txt

2^0*(2^1-1)+1 is trivially prime!: 2
2^1*(2^2-1)+1 is trivially prime!: 7
2^2*(2^3-1)+1 is trivially prime!: 29
2^4*(2^5-1)+1 trivially factors as: 7*71
2^6*(2^7-1)+1 trivially factors as: 11*739
2^12*(2^13-1)+1 is trivially prime!: 33550337
2^16*(2^17-1)+1 has factors: 7
2^18*(2^19-1)+1 is 3-PRP! (0.0000s+0.0009s)
2^30*(2^31-1)+1 has factors: 29
2^60*(2^61-1)+1 is composite: RES64: [36E090A8C361AD6C] (0.0000s+0.0003s)
2^88*(2^89-1)+1 has factors: 7
2^106*(2^107-1)+1 has factors: 7
2^126*(2^127-1)+1 has factors: 11
2^520*(2^521-1)+1 has factors: 7
2^606*(2^607-1)+1 has factors: 11
2^1278*(2^1279-1)+1 is composite: RES64: [570A6B3FD91E6339] (0.8700s+0.0011s)
2^2202*(2^2203-1)+1 is composite: RES64: [ECB4FE924C674723] (4.6906s+0.0010s)
2^2280*(2^2281-1)+1 has factors: 197
2^3216*(2^3217-1)+1 has factors: 11
2^4252*(2^4253-1)+1 has factors: 7
2^4422*(2^4423-1)+1 is composite: RES64: [F3603EEF4BD4F197] (17.0237s+0.0031s)
2^9688*(2^9689-1)+1 has factors: 7
2^9940*(2^9941-1)+1 has factors: 7
2^11212*(2^11213-1)+1 has factors: 7
2^19936*(2^19937-1)+1 has factors: 7
2^21700*(2^21701-1)+1 has factors: 7
2^23208*(2^23209-1)+1 has factors: 35603
2^44496*(2^44497-1)+1 has factors: 11
2^86242*(2^86243-1)+1 has factors: 7
2^110502*(2^110503-1)+1 has factors: 491
2^132048*(2^132049-1)+1 is composite: RES64: [1B3B60AEC3578817] (744.2790s+111.7145s)
2^216090*(2^216091-1)+1 has factors: 4673
2^756838*(2^756839-1)+1 has factors: 7
2^859432*(2^859433-1)+1 has factors: 7
2^1257786*(2^1257787-1)+1 has factors: 11
2^1398268*(2^1398269-1)+1 has factors: 7
2^2976220*(2^2976221-1)+1 has factors: 7
2^3021376*(2^3021377-1)+1 has factors: 7
2^6972592*(2^6972593-1)+1 has factors: 7
2^13466916*(2^13466917-1)+1 has factors: 11


The rest Mersenne primes lead to small factors:

Format ($p$,factor) (24036583,149),(25964951,7),( 30402457,11),( 32582657,7),( 37156667,7),( 42643801,3593),( 43112609,7)

2 Only one candidate left

For dealing with large potential primes a good choice is openpfgw

Using openpfgw I finished the list to $1 3466 917$ in about 20 minutes without finding new primes.

[added] The only prime perfect + 1 candidate from the known Mersenne primes is for $M_{20996011}$ - I am running ECM factoring on it.

Here is the log:

./pfgw64 -f10 -lmer1log.txt /tmp/mer.txt

2^0*(2^1-1)+1 is trivially prime!: 2
2^1*(2^2-1)+1 is trivially prime!: 7
2^2*(2^3-1)+1 is trivially prime!: 29
2^4*(2^5-1)+1 trivially factors as: 7*71
2^6*(2^7-1)+1 trivially factors as: 11*739
2^12*(2^13-1)+1 is trivially prime!: 33550337
2^16*(2^17-1)+1 has factors: 7
2^18*(2^19-1)+1 is 3-PRP! (0.0000s+0.0009s)
2^30*(2^31-1)+1 has factors: 29
2^60*(2^61-1)+1 is composite: RES64: [36E090A8C361AD6C] (0.0000s+0.0003s)
2^88*(2^89-1)+1 has factors: 7
2^106*(2^107-1)+1 has factors: 7
2^126*(2^127-1)+1 has factors: 11
2^520*(2^521-1)+1 has factors: 7
2^606*(2^607-1)+1 has factors: 11
2^1278*(2^1279-1)+1 is composite: RES64: [570A6B3FD91E6339] (0.8700s+0.0011s)
2^2202*(2^2203-1)+1 is composite: RES64: [ECB4FE924C674723] (4.6906s+0.0010s)
2^2280*(2^2281-1)+1 has factors: 197
2^3216*(2^3217-1)+1 has factors: 11
2^4252*(2^4253-1)+1 has factors: 7
2^4422*(2^4423-1)+1 is composite: RES64: [F3603EEF4BD4F197] (17.0237s+0.0031s)
2^9688*(2^9689-1)+1 has factors: 7
2^9940*(2^9941-1)+1 has factors: 7
2^11212*(2^11213-1)+1 has factors: 7
2^19936*(2^19937-1)+1 has factors: 7
2^21700*(2^21701-1)+1 has factors: 7
2^23208*(2^23209-1)+1 has factors: 35603
2^44496*(2^44497-1)+1 has factors: 11
2^86242*(2^86243-1)+1 has factors: 7
2^110502*(2^110503-1)+1 has factors: 491
2^132048*(2^132049-1)+1 is composite: RES64: [1B3B60AEC3578817] (744.2790s+111.7145s)
2^216090*(2^216091-1)+1 has factors: 4673
2^756838*(2^756839-1)+1 has factors: 7
2^859432*(2^859433-1)+1 has factors: 7
2^1257786*(2^1257787-1)+1 has factors: 11
2^1398268*(2^1398269-1)+1 has factors: 7
2^2976220*(2^2976221-1)+1 has factors: 7
2^3021376*(2^3021377-1)+1 has factors: 7
2^6972592*(2^6972593-1)+1 has factors: 7
2^13466916*(2^13466917-1)+1 has factors: 11


The rest Mersenne primes lead to small factors:

Format ($p$,factor) (24036583,149),(25964951,7),( 30402457,11),( 32582657,7),( 37156667,7),( 42643801,3593),( 43112609,7)

1

For dealing with large potential primes a good choice is openpfgw

Using openpfgw I finished the list to $1 3466 917$ in about 20 minutes without finding new primes.

Here is the log:

./pfgw64 -f10 -lmer1log.txt /tmp/mer.txt

2^0*(2^1-1)+1 is trivially prime!: 2
2^1*(2^2-1)+1 is trivially prime!: 7
2^2*(2^3-1)+1 is trivially prime!: 29
2^4*(2^5-1)+1 trivially factors as: 7*71
2^6*(2^7-1)+1 trivially factors as: 11*739
2^12*(2^13-1)+1 is trivially prime!: 33550337
2^16*(2^17-1)+1 has factors: 7
2^18*(2^19-1)+1 is 3-PRP! (0.0000s+0.0009s)
2^30*(2^31-1)+1 has factors: 29
2^60*(2^61-1)+1 is composite: RES64: [36E090A8C361AD6C] (0.0000s+0.0003s)
2^88*(2^89-1)+1 has factors: 7
2^106*(2^107-1)+1 has factors: 7
2^126*(2^127-1)+1 has factors: 11
2^520*(2^521-1)+1 has factors: 7
2^606*(2^607-1)+1 has factors: 11
2^1278*(2^1279-1)+1 is composite: RES64: [570A6B3FD91E6339] (0.8700s+0.0011s)
2^2202*(2^2203-1)+1 is composite: RES64: [ECB4FE924C674723] (4.6906s+0.0010s)
2^2280*(2^2281-1)+1 has factors: 197
2^3216*(2^3217-1)+1 has factors: 11
2^4252*(2^4253-1)+1 has factors: 7
2^4422*(2^4423-1)+1 is composite: RES64: [F3603EEF4BD4F197] (17.0237s+0.0031s)
2^9688*(2^9689-1)+1 has factors: 7
2^9940*(2^9941-1)+1 has factors: 7
2^11212*(2^11213-1)+1 has factors: 7
2^19936*(2^19937-1)+1 has factors: 7
2^21700*(2^21701-1)+1 has factors: 7
2^23208*(2^23209-1)+1 has factors: 35603
2^44496*(2^44497-1)+1 has factors: 11
2^86242*(2^86243-1)+1 has factors: 7
2^110502*(2^110503-1)+1 has factors: 491
2^132048*(2^132049-1)+1 is composite: RES64: [1B3B60AEC3578817] (744.2790s+111.7145s)
2^216090*(2^216091-1)+1 has factors: 4673
2^756838*(2^756839-1)+1 has factors: 7
2^859432*(2^859433-1)+1 has factors: 7
2^1257786*(2^1257787-1)+1 has factors: 11
2^1398268*(2^1398269-1)+1 has factors: 7
2^2976220*(2^2976221-1)+1 has factors: 7
2^3021376*(2^3021377-1)+1 has factors: 7
2^6972592*(2^6972593-1)+1 has factors: 7
2^13466916*(2^13466917-1)+1 has factors: 11