Smallest Titanics of Special Forms

This first list was first compiled by David Broadhurst

• The smallest pair of titanic primes is
  • 10^999+7
  • 10^999+663
  as proven by Preda Mihailescu and Giovanni La Barbera.
• The smallest pair of titanic twin primes is
  • 10^999+1975081
  • 10^999+1975083
  as proven by Heuer Daniel (11 Oct 1999).
• The smallest titanic Sophie Germain pair is
  • 10^999+2222239
  • 2*10^999+4444479
  as proven by Giovanni La Barbera.
• The smallest semi-titanic Sophie Germain pair is
  • 10^999+1043119
  • 5*10^998+521559
  as proven by Giovanni La Barbera.
• The smallest titanic 4-fold arithmetic progression is
  • 10^999+2059323
  • 10^999+2139213
  • 10^999+2219103
  • 10^999+2298993
  as proven by David Broadhurst.
• The smallest titanic 3-fold arithmetic progression is
  • 10^999+61971
  • 10^999+91737
  • 10^999+121503
  as proven by David Broadhurst.
• The smallest titanic Cunningham chain of 2nd kind is
  • 10^999+2209041
  • 2*10^999+4418081
  as proven by David Broadhurst.
• The smallest semi-titanic Cunningham chain of 2nd kind is
  • 10^999+547137
  • 5*10^998+273569
  as proven by David Broadhurst.

David Broadhurst's eleven primes above were proven by nine runs of Certifix. Minimality was proven by 1,149,496 PrimeForm tests.

Norman Luhn adds

• The smallest titanic triplet is
  • 10^999+1598241813
  • 10^999+1598241817
  • 10^999+1598241819
  as proven by Norman Luhn.
• The smallest titanic triplet with pattern d=0,2,6 is
  • 10^999+5537073001
  • 10^999+5537073003
  • 10^999+5537073007
  as proven by Norman Luhn.
• The smallest titanic quadruple is
  • 10^999+4114571944591
  • 10^999+4114571944593
  • 10^999+4114571944597
  • 10^999+4114571944599
  as proven by Norman Luhn.
• The smallest titanic quintuplet (has pattern d=0,2,6,8,12) is
  • 10^999+3554007760224751
  • 10^999+3554007760224753
  • 10^999+3554007760224757
  • 10^999+3554007760224759
  • 10^999+3554007760224763
  as proven by Norman Luhn (6/2014).
• The smallest titanic quintuplet with pattern d=0,4,6,10,12 is
  • 10^999+3818999670116007
  • 10^999+3818999670116011
  • 10^999+3818999670116013
  • 10^999+3818999670116017
  • 10^999+3818999670116019
  as proven by Norman Luhn (1/2020).
• Smallest cousin primes (n,n+4)
  • 10^999+5352807
  • 10^999+5352811
• Smallest sexy primes (n,n+6)
  • 10^999+73203
  • 10^999+73209
  as proven by Jim Fougeron.
• Smallest sexy prime triplet (n,n+6,n+12)
  • 10^999+4975694077
  • 10^999+4975694083
  • 10^999+4975694089
  as proven by Norman Luhn.
• Smallest sexy prime quadruplet (n,n+6,n+12,n+18)
  • 10^999+465621162291
  • 10^999+465621162297
  • 10^999+465621162303
  • 10^999+465621162309
  as proven by Jim Fougeron.

Dirk Augustin adds

• The smallest titanic Cunningham chain of length 3 starts with
  • 10^999+1964944441
  as proven by Dirk Augustin.
• The smallest titanic Cunningham chain, 2nd kind, of length 3 starts with
  • 10^999+12142617231
  as proven by Dirk Augustin.

Phil Carmody added this list of smallest titanic Generalized Fermat primes:

• n²+1 where n is

  31622776601 6837933199 8893544432 718533719 5551393252 1682685750 4852792594 4386392382 2134424810 8379300295 1873472841 5284005514 8548856030 4538800146 9051959670 0153903344 9216571792 5994065915 0153474113 3394841240 8531692957 7090471576 4610443692 5787906203 7808609941 8283717115 4840632855 2999118596 8245642033 2696160469 1314336128 9497918902 6652954361 2676178781 3500613881 8627858046 3683134952 4780311437 6933467197 3819513185 6784032312 4179540221 8308045872 8446146002 5357757970 2828644029 0244079778 9603454398 9163349222 6526121090

  (Proof: Primo 2)
• n⁴+1 where n is

  5623413251 9034908039 4951039776 4812314682 5104309869 1664081689 4237358835 6864306284 8905857984 5262203059 2867610732 0100325218 0092284975 7565578997 7624934608 1029794998 3883322661 3000142162 9615341734 1225320759 5084019528 0008234806 7853926543 8612654811 6667382338

  (Proof: Primo 2)
  
• n⁸+1 where n is

  74989 4209332455 8273021842 7561513643 8441867918 1649710146 2041900542 9827525167 1606279806 7369598314 4556246592 0840077240 5854520854

  (Proof: Primo 2)
  
• n¹⁶+1 where n is

  273 8419634264 3612941886 9698738915 8046926067 5814465384 7798184544

  (Proof: n-1, 100% factorization.)
• n³²+1 where n is

  16 5481709994 3181422945 6139402996

  (Proof: n-1, 100% factorization.)
  
• n⁶⁴+1 where n is 4067944321083186. (Proof: n-1, 100% factorization.)
  
• n¹²⁸+1 where n is 63780790. (Proof: n-1, 100% factorization.)