Wikipedia:Sandbox

Although these numbers also have *N*−1 or *N*+1 is product of a Cunningham number and a small number, but since the corresponding Cunningham numbers are < 25% factored, and the partial factorizations of them are insufficient for any of the proving methods that could make use of them (unless the second-greatest prime factor (https://oeis.org/A087039, https://stdkmd.net/nrr/records.htm#BIGFACTOR, https://stdkmd.net/nrr/repunit/changes202411.htm (table "Largest known factors that appear after the previous one" in section "November 18, 2024"), https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Largest penultimate prime factor (ultimate factor shown also):"), http://myfactorcollection.mooo.com:8090/ruminations.html (section "primitives, top 123 pen-ultimate factors")) of this number (*N*−1 or *N*+1) is < 10¹⁰⁰ (i.e. this number (*N*−1 or *N*+1) is product of a 10¹⁰⁰-smooth number (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) and a prime) and is thus accessible by massive *ECM* (or *P*−1 or *P*+1) computations, or the product of the prime factors > 10¹⁰⁰ of this number (*N*−1 or *N*+1) is < 10²⁵⁰ (i.e. this number (*N*−1 or *N*+1) is product of a 10¹⁰⁰-smooth number (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) and a number < 10²⁵⁰) and is thus accessible by combine of massive *ECM* (or *P*−1 or *P*+1) computations and *GNFS*, there is no chance for fully factoring this number (*N*−1 or *N*+1) using current publicly available factorization methods if the *SNFS* difficulty of this number (*N*−1 or *N*+1) is > 360, since the currently record of the *SNFS* factorization is difficulty 360 (which is the number 2¹¹⁹³−1, see https://homes.cerias.purdue.edu/~ssw/cun/champ.txt) and the currently record of the *GNFS* factorization number has 250 decimal digits (which is the number *RSA*-250, see https://en.wikipedia.org/wiki/Integer_factorization_records#Numbers_of_a_general_form) and the currently record of the *ECM* prime factor has 83 decimal digits (which is a prime factor of 7³³⁷+1, see http://www.loria.fr/~zimmerma/records/top50.html) and the currently record of the *P*−1 prime factor has 66 decimal digits (which is a prime factor of 960¹¹⁹−1, see http://www.loria.fr/~zimmerma/records/Pminus1.html) and the currently record of the *P*+1 prime factor has 60 decimal digits (which is a prime factor of the Lucas number *L*₂₃₆₆, see http://www.loria.fr/~zimmerma/records/Pplus1.html), thus there is no change to prove their primality using the *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) or the *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2)), like the numbers (13¹¹⁹³−1)/12 (see https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html and its *factordb* entry http://factordb.com/index.php?id=1000000000043597217&open=prime and its primality certificate http://factordb.com/cert.php?id=1000000000043597217 and its certificate chain http://factordb.com/certchain.php?fid=1000000000043597217&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1000000000043597217 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000271071123&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=1192&c0=-&EN=&LM=) and (55⁸³⁹−1)/54 (see https://web.archive.org/web/20020821230129/http://www.users.globalnet.co.uk/~aads/C0550839.html and its *factordb* entry http://factordb.com/index.php?id=1100000000672342180&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000672342180 and its certificate chain http://factordb.com/certchain.php?fid=1100000000672342180&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000672342180 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000674669599&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=55&Exp=838&c0=-&EN=&LM=) and (70¹⁰¹³−1)/69 (see https://web.archive.org/web/20020825072348/http://www.users.globalnet.co.uk/~aads/C0701013.html and its *factordb* entry http://factordb.com/index.php?id=1100000000599116446&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000599116446 and its certificate chain http://factordb.com/certchain.php?fid=1100000000599116446&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000599116446 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000599116447&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=70&Exp=1012&c0=-&EN=&LM=) and (79⁶⁵⁹−1)/78 (see https://web.archive.org/web/20020825073634/http://www.users.globalnet.co.uk/~aads/C0790659.html and its *factordb* entry http://factordb.com/index.php?id=1100000000235993821&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000235993821 and its certificate chain http://factordb.com/certchain.php?fid=1100000000235993821&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000235993821 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000271854142&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=79&Exp=658&c0=-&EN=&LM=) and (101⁶⁷⁷−1)/100 (see https://web.archive.org/web/20020827132442/http://www.users.globalnet.co.uk/~aads/C1010677.html and its *factordb* entry http://factordb.com/index.php?id=1100000000236024785&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000236024785 and its certificate chain http://factordb.com/certchain.php?fid=1100000000236024785&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000236024785 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000271853242&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=101&Exp=676&c0=-&EN=&LM=) and (123⁵⁶³−1)/122 (see https://web.archive.org/web/20020901180251/http://www.users.globalnet.co.uk/~aads/C1230563.html and its *factordb* entry http://factordb.com/index.php?id=1100000000672342354&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000672342354 and its certificate chain http://factordb.com/certchain.php?fid=1100000000672342354&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000672342354 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000672583800&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=123&Exp=562&c0=-&EN=&LM=) and (133⁵⁹⁹−1)/132 (see https://web.archive.org/web/20020906181726/http://www.users.globalnet.co.uk/~aads/C1330599.html and its *factordb* entry http://factordb.com/index.php?id=1100000000696698470&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000696698470 and its certificate chain http://factordb.com/certchain.php?fid=1100000000696698470&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000696698470 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000696778792&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=133&Exp=598&c0=-&EN=&LM=) and (10⁴⁹⁰⁸¹−1)/9 (see https://www.mersenneforum.org/showthread.php?t=13435 and its *factordb* entry http://factordb.com/index.php?id=1100000000013937242&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000013937242 and its certificate chain http://factordb.com/certchain.php?fid=1100000000013937242&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000013937242 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000020361525&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10&Exp=49080&c0=-&EN=&LM=) and (71¹⁶³⁸⁴+1)/2 (see section "Faktorisieren der Zahl (71^16384+1)/2-1" of http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt and its *factordb* entry http://factordb.com/index.php?id=1100000000213085670&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000213085670 and its certificate chain http://factordb.com/certchain.php?fid=1100000000213085670&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000213085670 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000710475165&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=71&Exp=16384&c0=-&EN=&LM=) and (28839⁸³¹⁷−1)/28838 (see https://web.archive.org/web/20131020162127/http://www.primes.viner-steward.org/andy/D/70A7207D.html and its *factordb* entry http://factordb.com/index.php?id=1100000000439222443&open=prime and its helper file http://factordb.com/helper.php?id=1100000000439222443 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000489615372&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=28839&Exp=8316&c0=-&EN=&LM= (this prime is still "PRP" in *factordb* and thus has no primality certificate and certificate chain since its *N*−1 is only 27.98% factored and thus need to use *CHG* proof, however, *factordb* lacks the ability to verify *CHG* proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165)) and (4366¹⁰⁰⁹⁹−1)/4365 (see https://web.archive.org/web/20131020160641/http://www.primes.viner-steward.org/andy/D/110E2773.html and its *factordb* entry http://factordb.com/index.php?id=1100000000439222455&open=prime and its helper file http://factordb.com/helper.php?id=1100000000439222455 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000489615383&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=4366&Exp=5049&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=4366&Exp=5049&c0=%2B&EN=&LM= (this prime is still "PRP" in *factordb* and thus has no primality certificate and certificate chain since its *N*−1 is only 32.068% factored and thus need to use *CHG* proof, however, *factordb* lacks the ability to verify *CHG* proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165), also use 4366⁵⁰⁴⁹−1 and 4366⁵⁰⁴⁹+1 instead of 4366¹⁰⁰⁹⁸−1 for http://myfactors.mooo.com/ (note that 4366¹⁰⁰⁹⁸−1 = (4366⁵⁰⁴⁹−1) × (4366⁵⁰⁴⁹+1)) since 4366¹⁰⁰⁹⁸−1 is beyond the table limit in http://myfactors.mooo.com/, the table limit for *b*^*n*±1 in http://myfactors.mooo.com/ is (only consider non-perfect power bases *b*, i.e. *b* is not of the form *m*^*r* with *r* > 1) is *n* ≤ 2000000 for base *b* = 2, *n* ≤ 600000 for base *b* = 3, *n* ≤ 400000 for bases 5 ≤ *b* ≤ 7, *n* ≤ 300000 for base *b* = 10, *n* ≤ 100000 for bases 11 ≤ *b* ≤ 99, *n* ≤ 10000 for bases 101 ≤ *b* ≤ 11000000, and the table entry limit for *b*^*n*±1 in http://myfactors.mooo.com/ is (only consider non-perfect power bases *b*) is *n* ≤ 2000000 for base *b* = 2, *n* ≤ 600000 for base *b* = 3, *n* ≤ 400000 for bases 5 ≤ *b* ≤ 7, *n* ≤ 300000 for base *b* = 10, *n* ≤ 100000 for bases 11 ≤ *b* ≤ 99, *n* ≤ 10000 for bases 101 ≤ *b* ≤ 9999, *n* ≤ 500 for bases 10001 ≤ *b* ≤ 20000, *n* ≤ 300 for bases 20001 ≤ *b* ≤ 1100000, see http://myfactorcollection.mooo.com:8090/news.html and http://myfactorcollection.mooo.com:8090/interactive.html and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=2&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=2&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=3&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=3&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=4&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=4&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=5&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=5&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=6&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=6&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=7&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=7&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=8&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=8&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=9&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=9&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=11&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=11&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=12&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=12&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=99&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=99&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=100&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=100&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=101&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=101&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=9999&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=9999&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10000&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10000&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10001&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10001&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=999999999&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=999999999&Exp=999999999&c0=%2B&EN=&LM=, for perfect power bases *b* = *m*^*r* with *r* > 1, the table limit and the table entry limit can be converted from their "ground bases" (https://oeis.org/A052410) *b* = *m*, the limit of *b* = *m*^*r* is the floor function (https://en.wikipedia.org/wiki/Floor_function, https://t5k.org/glossary/xpage/FloorFunction.html, https://www.rieselprime.de/ziki/Floor_function, https://mathworld.wolfram.com/FloorFunction.html) of "1/*r* of the limit of *b* = *m*" (although you cannot enter the numbers directly if *n* > 10000, you need to convert them yourself before enter these numbers if *n* > 10000), however, the table limit is 10000 for the perfect power bases *b* ≤ 11000000 such that the converting result is < 10000 (thus, in fact, for 101 ≤ *b* ≤ 3316 = *floor*(*sqrt*(11000000)), the even numbers *n* ≤ 20000 are allowed, but you must use (*b*²)^*n*/2 instead of *b*^*n* if *n* > 10000, also, for 101 ≤ *b* ≤ 222 = *floor*(*cbrt*(11000000)), the numbers *n* ≤ 30000 which are divisible by 3 are allowed, but you must use (*b*³)^*n*/3 instead of *b*^*n* if *n* > 10000), see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10201&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10201&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=1030301&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=1030301&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10941048&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10941048&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10995856&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10995856&Exp=999999999&c0=%2B&EN=&LM=) and (11379⁷⁴¹¹−1)/11378 (see https://web.archive.org/web/20131020162219/http://www.primes.viner-steward.org/andy/E/2C731CF3.html and its *factordb* entry http://factordb.com/index.php?id=1100000000439222463&open=prime and its helper file http://factordb.com/helper.php?id=1100000000439222463 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000489615394&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=11379&Exp=7410&c0=-&EN=&LM= (this prime is still "PRP" in *factordb* and thus has no primality certificate and certificate chain since its *N*−1 is only 30.835% factored and thus need to use *CHG* proof, however, *factordb* lacks the ability to verify *CHG* proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165)) and (13096⁵⁹⁵³−1)/13095 (see https://web.archive.org/web/20131020160719/http://www.primes.viner-steward.org/andy/E/33281741.html and its *factordb* entry http://factordb.com/index.php?id=1100000000439222497&open=prime and its helper file http://factordb.com/helper.php?id=1100000000439222497 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000489615397&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13096&Exp=5952&c0=-&EN=&LM= (this prime is still "PRP" in *factordb* and thus has no primality certificate and certificate chain since its *N*−1 is only 28.057% factored and thus need to use *CHG* proof, however, *factordb* lacks the ability to verify *CHG* proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165)) and 3^1681130+3⁴⁴⁵⁷⁸¹+1 (see http://csic.som.emory.edu/~lzhou/blogs/?p=717 (although this page misses the Aurifeuillean factorization of *Φ*_6×*n*(3) = *Φ*_(6×*n*)*L*(3) × *Φ*_(6×*n*)*M*(3) for odd *n*) and its *factordb* entry http://factordb.com/index.php?id=1100000003878235677&open=prime and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000006004342031&open=ecm (this prime is "U" in *factordb* and thus has no primality certificate and certificate chain and helper file since this prime is too large (>10¹⁹⁹⁹⁹⁹) to be PRP-tested in *factordb*, also there is no factorization status of its *N*−1 in http://myfactors.mooo.com/ (i.e. there is no factorization status of its 3^1235349+1 in http://myfactors.mooo.com/) since 3^1235349+1 is beyond the table limit in http://myfactors.mooo.com/, the table limit for *b*^*n*±1 in http://myfactors.mooo.com/ is (only consider non-perfect power bases *b*, i.e. *b* is not of the form *m*^*r* with *r* > 1) is *n* ≤ 2000000 for base *b* = 2, *n* ≤ 600000 for base *b* = 3, *n* ≤ 400000 for bases 5 ≤ *b* ≤ 7, *n* ≤ 300000 for base *b* = 10, *n* ≤ 100000 for bases 11 ≤ *b* ≤ 99, *n* ≤ 10000 for bases 101 ≤ *b* ≤ 11000000, and the table entry limit for *b*^*n*±1 in http://myfactors.mooo.com/ is (only consider non-perfect power bases *b*) is *n* ≤ 2000000 for base *b* = 2, *n* ≤ 600000 for base *b* = 3, *n* ≤ 400000 for bases 5 ≤ *b* ≤ 7, *n* ≤ 300000 for base *b* = 10, *n* ≤ 100000 for bases 11 ≤ *b* ≤ 99, *n* ≤ 10000 for bases 101 ≤ *b* ≤ 9999, *n* ≤ 500 for bases 10001 ≤ *b* ≤ 20000, *n* ≤ 300 for bases 20001 ≤ *b* ≤ 1100000, see http://myfactorcollection.mooo.com:8090/news.html and http://myfactorcollection.mooo.com:8090/interactive.html and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=2&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=2&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=3&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=3&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=4&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=4&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=5&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=5&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=6&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=6&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=7&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=7&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=8&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=8&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=9&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=9&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=11&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=11&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=12&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=12&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=99&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=99&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=100&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=100&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=101&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=101&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=9999&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=9999&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10000&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10000&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10001&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10001&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=999999999&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=999999999&Exp=999999999&c0=%2B&EN=&LM=, for perfect power bases *b* = *m*^*r* with *r* > 1, the table limit and the table entry limit can be converted from their "ground bases" (https://oeis.org/A052410) *b* = *m*, the limit of *b* = *m*^*r* is the floor function (https://en.wikipedia.org/wiki/Floor_function, https://t5k.org/glossary/xpage/FloorFunction.html, https://www.rieselprime.de/ziki/Floor_function, https://mathworld.wolfram.com/FloorFunction.html) of "1/*r* of the limit of *b* = *m*" (although you cannot enter the numbers directly if *n* > 10000, you need to convert them yourself before enter these numbers if *n* > 10000), however, the table limit is 10000 for the perfect power bases *b* ≤ 11000000 such that the converting result is < 10000 (thus, in fact, for 101 ≤ *b* ≤ 3316 = *floor*(*sqrt*(11000000)), the even numbers *n* ≤ 20000 are allowed, but you must use (*b*²)^*n*/2 instead of *b*^*n* if *n* > 10000, also, for 101 ≤ *b* ≤ 222 = *floor*(*cbrt*(11000000)), the numbers *n* ≤ 30000 which are divisible by 3 are allowed, but you must use (*b*³)^*n*/3 instead of *b*^*n* if *n* > 10000), see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10201&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10201&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=1030301&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=1030301&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10941048&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10941048&Exp=999999999&c0=%2B&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10995856&Exp=999999999&c0=-&EN=&LM= and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10995856&Exp=999999999&c0=%2B&EN=&LM=)), for more examples see https://web.archive.org/web/20240305200806/https://stdkmd.net/nrr/prime/primesize.txt and https://web.archive.org/web/20240305201054/https://stdkmd.net/nrr/prime/primesize.zip (see which numbers have "-" or "+" in the "note" column), thus we treat these numbers as integers with no special form (i.e. ordinary primes (https://t5k.org/glossary/xpage/OrdinaryPrime.html)) and prove its primality with *Primo* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309), and these numbers still need primality certificates: