Wikipedia:Sandbox

|family|corresponding form
({*x*} or *x*{*y*} or {*x*}*y* or *x*{0}*y*)|the value of *x*|the value of *y*|smallest allowed *b*|smallest allowed *n*|*OEIS* sequences for the smallest *n* such that this form is prime for fixed base *b* (such *n* always exist unless these families can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them) if my conjecture is true)|*OEIS* sequences for the smallest base *b* such that this form is prime for fixed *n* (such base *b* always exist unless these families can be ruled out as only containing composites (by single prime factor or algebraic factorization) if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true, in fact, if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true, then all numbers not in the *OEIS* sequence https://oeis.org/A121719 are primes in infinitely many bases *b*, since if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true, then all irreducible polynomials (https://en.wikipedia.org/wiki/Irreducible_polynomial, https://mathworld.wolfram.com/IrreduciblePolynomial.html) *a*_*n**x*^*n*+*a*_*n*−1*x*^*n*−1+*a*_*n*−2*x*^*n*−2+...+*a*₂*x*²+*a*₁*x*+*a*₀ which have no fixed prime factors (in fact, such prime factors must be ≤ *n*, i.e. ≤ the degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) of the polynomial) for all integers *x* contain infinitely many primes, see https://oeis.org/A354718 and https://oeis.org/A337164)
(although these primes need not to be minimal primes in base *b*, I include this only because these *OEIS* sequences are usable references of the primes in these families)|references|current smallest base *b* such that this family is an unsolved family (i.e. have no known prime (or strong probable prime) members > *b*, nor can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them))|search limit of the length of this family in this base *b*|bases *b* such that this family can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them)
bases *b*: why this family contains no primes > *b*
(only list reasons such that there are bases 2 ≤ *b* ≤ 2048 which the reason is realized)|smaller bases *b* with the smallest (probable) prime in this family has length > 100: *b* (*length*)|

|---|---|---|---|---|---|---|---|---|---|---|---|---|
|(*b**n*−1)/(*b*−1)|{*x*}|1|–|2|2|https://oeis.org/A084740
https://oeis.org/A084738 (corresponding primes)
https://oeis.org/A246005 (odd *b*)
https://oeis.org/A065854 (prime *b*)
https://oeis.org/A279068 (prime *b*, corresponding primes)
https://oeis.org/A360738 (*n* replaced by *n*−1)
https://oeis.org/A279069 (prime *b*, *n* replaced by *n*−1)
https://oeis.org/A128164 (*n* = 2 not allowed)
https://oeis.org/A285642 (*n* = 2 not allowed, corresponding primes)
https://oeis.org/A065813 (prime *b*, *n* = 2 not allowed, *n* replaced by (*n*−1)/2)|https://oeis.org/A066180
https://oeis.org/A084732 (corresponding primes)
**(if this form is prime, then *n* must be a prime, see https://t5k.org/notes/proofs/Theorem2.html for the proof, this proof can be generalized to any base *b*, see https://en.wikipedia.org/wiki/Repunit#Properties)**%7Chttp://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German)
https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html
http://www.primenumbers.net/Henri/us/MersFermus.htm
https://sites.google.com/view/repunit-and-antirepunit
http://www.bitman.name/math/table/379 (in Italian)
https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf)
https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf)%7C185%7C350000%7C*b* = *m*2: difference-of-two-squares factorization
*b* = *m*3: difference-of-two-cubes factorization
*b* = *m*5: difference-of-two-5th-powers factorization
*b* = *m*7: difference-of-two-7th-powers factorization
**(note: although bases *b* = 4, 8, 16, 27, 36, 100, 128 have algebraic factorization (if the numbers are factored as *F* × *G* / *d*), *F* (or *G*) is equal to *d* and *G* (or *F*) is prime > *b*, to make the factorizations be trivial, thus these bases *b* have only one very small prime > *b* instead of "can be ruled out as only containing composites (only count the numbers > *b*)", thus the only smaller bases *b* such that this family can be ruled out as only containing composites (only count the numbers > *b*) are 9, 25, 32, 49, 64, 81, 121, 125, 144, 169)**|35 (313)
39 (349)
47 (127)
51 (4229)
91 (4421)
92 (439)
124 (599)
135 (1171)
139 (163)
142 (1231)
152 (270217)
171 (181)
174 (3251)
182 (167)
183 (223)
184 (16703)|
|*b**n*+1|*x*{0}*y*|1|1|2|1|https://oeis.org/A079706
https://oeis.org/A084712 (corresponding primes)
https://oeis.org/A228101 (*n* replaced by *log*2*n*)
https://oeis.org/A123669 (*n* = 1 not allowed, corresponding primes)|https://oeis.org/A056993
https://oeis.org/A123599 (corresponding primes)
**(if this form is prime, then *n* must be a power of 2, see https://web.archive.org/web/20231001191526/http://yves.gallot.pagesperso-orange.fr/primes/math.html for the proof, this proof can be generalized to any base *b*, see https://www.mersenneforum.org/showpost.php?p=95745&postcount=3 and https://www.mersenneforum.org/showpost.php?p=96001&postcount=95)**%7Chttp://jeppesn.dk/generalized-fermat.html
http://www.noprimeleftbehind.net/crus/GFN-primes.htm
https://sites.google.com/view/generalized-fermat-primes
https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html
https://web.archive.org/web/20231003030159/http://yves.gallot.pagesperso-orange.fr/primes/results.html
https://web.archive.org/web/20231001191355/http://yves.gallot.pagesperso-orange.fr/primes/stat.html%7C38%7C33554432%7C*b* == 1 mod 2: always divisible by 2
*b* = *m*3: sum-of-two-cubes factorization
*b* = *m*5: sum-of-two-5th-powers factorization|(none)|
|(*b**n*+1)/2|{*x*}*y*|(*b*−1)/2|(*b*+1)/2|3
(only odd *b*)|2||https://oeis.org/A275530
**(if this form is prime, then *n* must be a power of 2, see https://web.archive.org/web/20231001191526/http://yves.gallot.pagesperso-orange.fr/primes/math.html for the proof, this proof can be generalized to any base *b*, see https://www.mersenneforum.org/showpost.php?p=95745&postcount=3 and https://www.mersenneforum.org/showpost.php?p=96001&postcount=95)**%7Chttp://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (in German)
https://sites.google.com/view/generalized-fermat-primes%7C31%7C16777215%7C*b* = *m*3: sum-of-two-cubes factorization|(none)|
|2×*b**n*+1|*x*{0}*y*|2|1|3|1|https://oeis.org/A119624
https://oeis.org/A253178 (only bases *b* which have possible primes)
https://oeis.org/A098872 (*b* divisible by 6)||https://www.mersenneforum.org/showthread.php?t=6918
https://www.mersenneforum.org/showthread.php?t=19725 (*b* == 11 mod 12)
https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C365%7C500000%7C*b* == 1 mod 3: always divisible by 3|38 (2730)
47 (176)
101 (192276)
104 (1234)
117 (287)
122 (756)
137 (328)
147 (155)
167 (6548)
203 (106)
206 (46206)
218 (333926)
236 (161230)
248 (322)
257 (12184)
263 (958)
287 (5468)
305 (16808)
347 (124)
353 (2314)|
|2×*b**n*−1|*x*{*y*}|1|*b*−1|3|1|https://oeis.org/A119591
https://oeis.org/A098873 (*b* divisible by 6)
https://oeis.org/A279095 (power-of-2 *b*)|https://oeis.org/A157922%7Chttps://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217
https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C581%7C600000%7C(none)%7C29 (137)
67 (769)
74 (133)
107 (21911)
152 (797)
161 (229)
170 (166429)
191 (971)
215 (1073)
224 (109)
233 (8621)
235 (181)
254 (2867)
260 (121)
276 (2485)
278 (43909)
284 (417)
298 (4203)
303 (40175)
308 (991)
347 (523)
380 (3787)
382 (2325)
383 (20957)
393 (108)
395 (397)
401 (113)
418 (472)
422 (541)
431 (529)
434 (1167)
449 (175)
457 (103)
473 (661)
480 (145)
503 (861)
513 (299)
515 (58467)
522 (62289)
524 (165)
536 (841)
550 (1381)
551 (2719)
572 (3805)
578 (129469)|
|*b**n*+2|*x*{0}*y*|1|2|3|1|https://oeis.org/A138066
https://oeis.org/A084713 (corresponding primes)
https://oeis.org/A138067 (*n* = 1 not allowed)|https://oeis.org/A087576
https://oeis.org/A095302 (corresponding primes)||167|100000|*b* == 0 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3
*b* = 2*r* such that the equation 2*x* == −1 mod *r* has no solution but *r* is odd: combine of sum-of-two-*p*th-powers factorization for infinitely many odd primes *p* ((2*r*)*n*+2 = 2×(2*n*×*r*−1+1), and if 2*n*×*r*−1+1 has no algebraic factorization, then *n*×*r*−1 must be a power of 2 (otherwise, if *n*×*r*−1 has an odd prime factor *p*, then 2*n*×*r*−1+1 has a sum-of-two-*p*th-powers factorization), and this power of 2 must be == −1 mod *r*) (for all such *r* see https://oeis.org/A014659, and for such *r* which are primes see https://oeis.org/A014663, these primes *r* are exactly the primes *r* such that *ord**r*(2) is odd, and the primitive elements of this sequence (i.e. numbers which are in this sequence, but none of their proper divisors are in this sequence) are 7, 15, 23, 31, 39, 47, 51, 55, 71, 73, 79, 85, 87, 89, 95, 103, 111, 123, 127, 143, 151, 159, 167, 183, 187, 191, 199, 215, 221, 223, 233, 239, 247, 263, 271, 291, 295, 303, 311, 319, 323, 327, 335, 337, 339, 359, 367, 383, 407, 411, 415, 431, 439, 447, 451, 463, 471, 479, 485, 487, 493, 503, 519, 535, 543, 551, 559, 579, 583, 591, 599, 601, 607, 629, 631, 647, 655, 671, 687, 695, 697, 703, 719, 723, 727, 731, 743, 751, 767, 771, 779, 807, 815, 823, 831, 839, 863, 871, 879, 881, 887, 895, 901, 911, 919, 937, 939, 951, 965, 967, 983, 991, 1003, 1007, ... (unfortunately this sequence is not in *OEIS*)) (they are in fact combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p*, for such *r* which are primes, it is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* such that *ord**r*(*p*) is even, e.g. the case of *b* = 128 (i.e. *r* = 7) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 32768 (i.e. *r* = 15) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 2097152 (i.e. *r* = 21) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 3 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 3) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 3); and the case of *b* = 8388608 (i.e. *r* = 23) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 23 (i.e. the odd primes *p* == 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 mod 23) (i.e. the odd primes *p* in https://oeis.org/A191065); and the case of *b* = 2147483648 (i.e. *r* = 31) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067); and the case of *b* = 34359738368 (i.e. *r* = 35) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 5 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 5) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 5); and the case of *b* = 549755813888 (i.e. *r* = 39) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 13 but not both (i.e. the odd primes *p* == 7, 14, 17, 19, 23, 28, 29, 31, 34, 35, 37, 38 mod 39) (i.e. the odd primes *p* in https://oeis.org/A191070); and the case of *b* = 35184372088832 (i.e. *r* = 45) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 140737488355328 (i.e. *r* = 47) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 47 (i.e. the odd primes *p* == 5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39 mod 47) (i.e. the odd primes *p* in https://oeis.org/A191072); and the case of *b* = 562949953421312 (i.e. *r* = 49) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, https://t5k.org/notes/Dirichlet.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes))|47 (114)
89 (256)
159 (137)|
|*b**n*−2|{*x*}*y*|*b*−1|*b*−2|3|2|https://oeis.org/A250200
https://oeis.org/A255707 (*n* = 1 allowed)
https://oeis.org/A084714 (*n* = 1 allowed, corresponding primes)
https://oeis.org/A292201 (prime *b*, *n* = 1 allowed)|https://oeis.org/A095303
https://oeis.org/A095304 (corresponding primes)|https://www.primepuzzles.net/puzzles/puzz_887.htm (*n* = 1 allowed)|305|30000|*b* == 0 mod 2: always divisible by 2|81 (130)
97 (747)
197 (164)
209 (126)
215 (134)
221 (552)
287 (3410)|
|3×*b**n*+1|*x*{0}*y*|3|1|4|1|https://oeis.org/A098877 (*b* divisible by 6)||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C718%7C850000%7C*b* == 1 mod 2: always divisible by 2|108 (271)
314 (281)
358 (9561)
386 (184)
424 (1106)
458 (108)
492 (157)
636 (142)
646 (159)
648 (647)
652 (621)
654 (217)
690 (358)|
|3×*b**n*−1|*x*{*y*}|2|*b*−1|4|1|https://oeis.org/A098876 (*b* divisible by 6)||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C588%7C500000%7C*b* == 1 mod 2: always divisible by 2|42 (2524)
202 (263)
212 (283)
238 (105)
422 (191)
432 (16003)
446 (4851)
452 (335)
464 (219)
522 (347)
532 (136)
572 (377)
582 (445)|
|*b**n*+3|*x*{0}*y*|1|3|4|1||https://oeis.org/A087577%7C%7C718%7C10000%7C*b* == 1 mod 2: always divisible by 2
*b* == 0 mod 3: always divisible by 3
*b* = 2*r* such that the equation 2*x* == −1 mod *r* has no solution but *r* is odd: combine of sum-of-two-*p*th-powers factorization for infinitely many odd primes *p* ((3*r*)*n*+3 = 6×(3*n*×*r*−1+1)/2, and if (3*n*×*r*−1+1)/2 has no algebraic factorization, then *n*×*r*−1 must be a power of 2 (otherwise, if *n*×*r*−1 has an odd prime factor *p*, then (3*n*×*r*−1+1)/2 has a sum-of-two-*p*th-powers factorization), and this power of 2 must be == −1 mod *r*) (for all such *r* see https://oeis.org/A014659, and for such *r* which are primes see https://oeis.org/A014663, these primes *r* are exactly the primes *r* such that *ord**r*(2) is odd, and the primitive elements of this sequence (i.e. numbers which are in this sequence, but none of their proper divisors are in this sequence) are 7, 15, 23, 31, 39, 47, 51, 55, 71, 73, 79, 85, 87, 89, 95, 103, 111, 123, 127, 143, 151, 159, 167, 183, 187, 191, 199, 215, 221, 223, 233, 239, 247, 263, 271, 291, 295, 303, 311, 319, 323, 327, 335, 337, 339, 359, 367, 383, 407, 411, 415, 431, 439, 447, 451, 463, 471, 479, 485, 487, 493, 503, 519, 535, 543, 551, 559, 579, 583, 591, 599, 601, 607, 629, 631, 647, 655, 671, 687, 695, 697, 703, 719, 723, 727, 731, 743, 751, 767, 771, 779, 807, 815, 823, 831, 839, 863, 871, 879, 881, 887, 895, 901, 911, 919, 937, 939, 951, 965, 967, 983, 991, 1003, 1007, ... (unfortunately this sequence is not in *OEIS*)) (they are in fact combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p*, for such *r* which are primes, it is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* such that *ord**r*(*p*) is even, e.g. the case of *b* = 2187 (i.e. *r* = 7) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 14348907 (i.e. *r* = 15) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 10460353203 (i.e. *r* = 21) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 3 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 3) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 3); and the case of *b* = 94143178827 (i.e. *r* = 23) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 23 (i.e. the odd primes *p* == 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 mod 23) (i.e. the odd primes *p* in https://oeis.org/A191065); and the case of *b* = 617673396283947 (i.e. *r* = 31) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067); and the case of *b* = 50031545098999707 (i.e. *r* = 35) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 5 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 5) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 5); and the case of *b* = 4052555153018976267 (i.e. *r* = 39) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 13 but not both (i.e. the odd primes *p* == 7, 14, 17, 19, 23, 28, 29, 31, 34, 35, 37, 38 mod 39) (i.e. the odd primes *p* in https://oeis.org/A191070); and the case of *b* = 2954312706550833698643 (i.e. *r* = 45) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 26588814358957503287787 (i.e. *r* = 47) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 47 (i.e. the odd primes *p* == 5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39 mod 47) (i.e. the odd primes *p* in https://oeis.org/A191072); and the case of *b* = 239299329230617529590083 (i.e. *r* = 49) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, https://t5k.org/notes/Dirichlet.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes))|382 (256)
388 (109)
412 (137)
530 (1399)
548 (118)
646 (9314)|
|*b**n*−3|{*x*}*y*|*b*−1|*b*−3|4|2||||1192|6000|*b* == 1 mod 2: always divisible by 2
*b* == 0 mod 3: always divisible by 3|52 (105)
94 (204)
152 (346)
154 (396)
290 (111)
302 (1061)
478 (1410)
512 (1600)
542 (1944)
676 (141)
698 (306)
754 (120)
760 (120)
1000 (330)
1006 (124)
1010 (226)
1022 (102)
1094 (1508)
1096 (135)|
|4×*b**n*+1|*x*{0}*y*|4|1|5|1||**(such base *b* does not exist if *n* is divisible by 4 because of the Aurifeuillean factorization of *x*4+4×*y*4)**|https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C32%7C1717986918%7C*b* == 1 mod 5: always divisible by 5
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = *m*4: Aurifeuillean factorization of *x*4+4×*y*4|23 (343)|
|4×*b**n*−1|*x*{*y*}|3|*b*−1|5|1||**(such base *b* does not exist if *n* is even because of the difference-of-two-squares factorization)**|https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C275%7C1000000%7C*b* == 1 mod 3: always divisible by 3
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = *m*2: difference-of-two-squares factorization
*b* == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization|47 (1556)
72 (1119850)
107 (252)
167 (1866)
212 (34414)
218 (23050)
236 (940)
240 (1402)
251 (272)
261 (820)
270 (89662)|
|*b**n*+4|*x*{0}*y*|1|4|5|1||**(such base *b* does not exist if *n* is divisible by 4 because of the Aurifeuillean factorization of *x*4+4×*y*4)**||139|18000|*b* == 0 mod 2: always divisible by 2
*b* == 1 mod 5: always divisible by 5
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = *m*4: Aurifeuillean factorization of *x*4+4×*y*4
*b* = 2*r* such that the equation 2*x* == −2 mod *r* has no solution but *r* is odd: combine of sum-of-two-*p*th-powers factorization for infinitely many odd primes *p* ((2*r*)*n*+4 = 4×(2*n*×*r*−2+1), and if 2*n*×*r*−2+1 has no algebraic factorization, then *n*×*r*−2 must be a power of 2 (otherwise, if *n*×*r*−2 has an odd prime factor *p*, then 2*n*×*r*−2+1 has a sum-of-two-*p*th-powers factorization), and this power of 2 must be == −2 mod *r*) (for all such *r* see https://oeis.org/A014659, and for such *r* which are primes see https://oeis.org/A014663, these primes *r* are exactly the primes *r* such that *ord**r*(2) is odd, and the primitive elements of this sequence (i.e. numbers which are in this sequence, but none of their proper divisors are in this sequence) are 7, 15, 23, 31, 39, 47, 51, 55, 71, 73, 79, 85, 87, 89, 95, 103, 111, 123, 127, 143, 151, 159, 167, 183, 187, 191, 199, 215, 221, 223, 233, 239, 247, 263, 271, 291, 295, 303, 311, 319, 323, 327, 335, 337, 339, 359, 367, 383, 407, 411, 415, 431, 439, 447, 451, 463, 471, 479, 485, 487, 493, 503, 519, 535, 543, 551, 559, 579, 583, 591, 599, 601, 607, 629, 631, 647, 655, 671, 687, 695, 697, 703, 719, 723, 727, 731, 743, 751, 767, 771, 779, 807, 815, 823, 831, 839, 863, 871, 879, 881, 887, 895, 901, 911, 919, 937, 939, 951, 965, 967, 983, 991, 1003, 1007, ... (unfortunately this sequence is not in *OEIS*)) (they are in fact combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p*, for such *r* which are primes, it is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* such that *ord**r*(*p*) is even, e.g. the case of *b* = 128 (i.e. *r* = 7) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 32768 (i.e. *r* = 15) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 2097152 (i.e. *r* = 21) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 3 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 3) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 3); and the case of *b* = 8388608 (i.e. *r* = 23) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 23 (i.e. the odd primes *p* == 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 mod 23) (i.e. the odd primes *p* in https://oeis.org/A191065); and the case of *b* = 2147483648 (i.e. *r* = 31) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067); and the case of *b* = 34359738368 (i.e. *r* = 35) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 5 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 5) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 5); and the case of *b* = 549755813888 (i.e. *r* = 39) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 13 but not both (i.e. the odd primes *p* == 7, 14, 17, 19, 23, 28, 29, 31, 34, 35, 37, 38 mod 39) (i.e. the odd primes *p* in https://oeis.org/A191070); and the case of *b* = 35184372088832 (i.e. *r* = 45) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 3 or 5 but not both (i.e. the odd primes *p* == 7, 11, 13, 14 mod 15) (i.e. the odd primes *p* in https://oeis.org/A191062); and the case of *b* = 140737488355328 (i.e. *r* = 47) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 47 (i.e. the odd primes *p* == 5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39 mod 47) (i.e. the odd primes *p* in https://oeis.org/A191072); and the case of *b* = 562949953421312 (i.e. *r* = 49) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, https://t5k.org/notes/Dirichlet.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes))|53 (13403)
113 (10647)|
|*b**n*−4|{*x*}*y*|*b*−1|*b*−4|5|2||**(such base *b* does not exist if *n* is even because of the difference-of-two-squares factorization)**||207|12000|*b* == 0 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = *m*2: difference-of-two-squares factorization
*b* == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization|65 (175)
93 (105)
123 (299)
135 (165)
137 (147)
141 (395)
173 (135)
183 (113)
191 (319)
203 (107)|
|5×*b**n*+1|*x*{0}*y*|5|1|6|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C512%7C1000000%7C*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3|122 (136)
170 (176)
200 (768)
248 (262)
266 (510)
308 (309756)
318 (127)
326 (400786)
332 (106)
350 (20392)
356 (596)
368 (208)
392 (152)
410 (108)
440 (826)|
|5×*b**n*−1|*x*{*y*}|4|*b*−1|6|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C338%7C300000%7C*b* == 1 mod 2: always divisible by 2|14 (19699)
68 (13575)
112 (133)
116 (157)
196 (9850)
206 (109)
254 (15451)
320 (233)|
|6×*b**n*+1|*x*{0}*y*|6|1|7|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C212%7C1750000%7C*b* == 1 mod 7: always divisible by 7
*b* == 34 mod 35: always divisible by some element of {5,7}|53 (144)
67 (4533)
93 (521)
108 (16318)
129 (16797)
144 (783)
163 (1304)
185 (171)
193 (149)|
|6×*b**n*−1|*x*{*y*}|5|*b*−1|7|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C234%7C1000000%7C*b* == 1 mod 5: always divisible by 5
*b* == 34 mod 35: always divisible by some element of {5,7}
*b* = 6×*m*2 with *m* == 2, 3 mod 5: combine of factor 5 and difference-of-two-squares factorization|48 (295)
118 (211)
119 (666)
154 (1990)
178 (119)
188 (951)|
|7×*b**n*+1|*x*{0}*y*|7|1|8|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C1136%7C10000%7C*b* == 1 mod 2: always divisible by 2|50 (517)
62 (309)
170 (179)
194 (281)
224 (689)
236 (347)
308 (107)
338 (793)
380 (475)
382 (519)
386 (121)
398 (17473)
434 (321)
466 (181)
500 (1997)
520 (198)
522 (235)
524 (127)
598 (423)
632 (8447)
638 (265)
644 (3379)
652 (185)
674 (181)
682 (796)
724 (388)
734 (189)
764 (189)
836 (5701)
868 (274)
892 (157)
920 (491)
926 (523)
930 (218)
958 (169)
960 (128)
974 (1589)
982 (313)
1004 (54849)
1082 (2113)
1102 (820)|
|7×*b**n*−1|*x*{*y*}|6|*b*−1|8|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C308%7C300000%7C*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3|68 (25396)
182 (210)
198 (117)
248 (3180)
260 (826)|
|8×*b**n*+1|*x*{0}*y*|8|1|9|1||**(such base *b* does not exist if *n* is divisible by 3 because of the sum-of-two-cubes factorization)**|https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C86%7C1000000%7C*b* == 1 mod 3: always divisible by 3
*b* == 20 mod 21: always divisible by some element of {3,7}
*b* == 47, 83 mod 195: always divisible by some element of {3,5,13}
*b* == 467, 4343, 9887, 25448, 35978, 41522, 42647, 57083 mod 73815: always divisible by some element of {3,5,7,19,37}
*b* == 722, 83813, 206672, 239432, 322523, 1283843, 1519577, 1522553 mod 1551615: always divisible by some element of {3,5,13,73,109}
*b* = *m*3: sum-of-two-cubes factorization
*b* = 2*r* such that the equation 2*x* == 3 mod *r* has no solution but *r* is not divisible by either 2 or 3: combine of sum-of-two-*p*th-powers factorization for infinitely many odd primes *p* (8×(2*r*)*n*+1 = 2*n*×*r*+3+1, and if 2*n*×*r*+3+1 has no algebraic factorization, then *n*×*r*+3 must be a power of 2 (otherwise, if *n*×*r*+3 has an odd prime factor *p*, then 2*n*×*r*+3+1 has a sum-of-two-*p*th-powers factorization), and this power of 2 must be == 3 mod *r*) (for such *r* which are primes see https://oeis.org/A123988, unfortunately there is no *OEIS* sequence for all such *r* or when "*r* is not divisible by either 2 or 3" is not required, nor the primitive elements of these sequences (i.e. numbers which are in these sequences, but none of their proper divisors are in these sequences) (such *r* are 7, 17, 31, 35, 41, 43, 49, 55, 65, 73, 77, 79, 85, 89, 91, 103, 109, 113, 119, 127, 133, 137, 145, 151, 155, 157, 161, 175, 185, 187, 199, 203, 205, 209, 215, 217, 221, 223, 229, 233, 241, 245, 247, 251, 257, 259, 265, 271, 275, 277, 281, 283, 287, 289, 295, 301, 305, 319, 323, 325, 329, 331, 337, 341, 343, 353, 365, 367, 371, 377, 385, 391, 395, 397, 401, 403, 413, 415, 425, 427, 433, 439, 445, 449, 451, 455, 457, 463, 469, 473, 481, 487, 493, 497, 505, 511, 515, 521, 527, 533, 535, 539, 545, 553, 559, 565, 569, 571, 581, 583, 589, 593, 595, 601, 605, 607, 617, 623, 629, 631, 635, 637, 641, 655, 665, 671, 673, 679, 683, 685, 689, 691, 697, 703, 707, 713, 715, 721, 725, 727, 731, 733, 737, 739, 749, 751, 755, 761, 763, 775, 779, 781, 785, 791, 793, 799, 803, 805, 809, 811, 817, 823, 833, 845, 847, 857, 869, 871, 875, 881, 889, 895, 899, 901, 905, 911, 917, 919, 925, 929, 931, 935, 937, 943, 949, 953, 959, 961, 965, 967, 971, 973, 977, 979, 985, 989, 991, 995, 1001, 1003, 1013, 1015, ..., and the primitive elements of this sequence (i.e. numbers which are in this sequence, but none of their proper divisors are in this sequence) are 7, 17, 31, 41, 43, 55, 65, 73, 79, 89, 103, 109, 113, 127, 137, 145, 151, 157, 185, 199, 209, 223, 229, 233, 241, 247, 251, 257, 265, 271, 277, 281, 283, 295, 305, 319, 331, 337, 353, 367, 377, 397, 401, 415, 433, 439, 449, 457, 463, 481, 487, 505, 521, 535, 569, 571, 583, 593, 601, 607, 617, 631, 641, 655, 671, 673, 683, 689, 691, 703, 727, 733, 737, 739, 751, 761, 781, 793, 809, 811, 823, 857, 871, 881, 895, 905, 911, 919, 929, 937, 953, 965, 967, 971, 977, 985, 991, 1013, ...) (they are in fact combine of sum-of-two-*p*th-powers factorization for *infinitely many* odd primes *p*, for such *r* which are primes, it is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not *q**s*th power residue (we only need consider the prime powers (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) *q**s* dividing *r*−1, for *q**s* = 2 this is quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html), for *q**s* = 3 this is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html), for *q**s* = 4 this is quartic residue (https://en.wikipedia.org/wiki/Quartic_reciprocity, https://mathworld.wolfram.com/BiquadraticResidue.html), for *q**s* = 8 this is octic residue (https://en.wikipedia.org/wiki/Octic_reciprocity), for other *q**s* see power residue symbol (https://en.wikipedia.org/wiki/Power_residue_symbol) and Dirichlet character (https://en.wikipedia.org/wiki/Dirichlet_character, https://mathworld.wolfram.com/NumberTheoreticCharacter.html, https://www.lmfdb.org/Character/Dirichlet/) and Eisenstein reciprocity (https://en.wikipedia.org/wiki/Eisenstein_reciprocity) and Artin reciprocity (https://en.wikipedia.org/wiki/Artin_reciprocity, https://mathworld.wolfram.com/ArtinsReciprocityTheorem.html)) mod *r* for all prime powers (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) *q**s* such that *q**s* divides https://oeis.org/A001917 at the entry of the prime *r* but *q**s* does not divide https://oeis.org/A094593 at the entry of the prime *r* but *q**s*−1 divides https://oeis.org/A094593 at the entry of the prime *r* (for prime *r*, 2*x* == 3 mod *r* has no solution is because *ord**r*(3) does not divide *ord**r*(2), i.e. https://oeis.org/A062117 at the entry of the prime *r* does not divide https://oeis.org/A014664 at the entry of the prime *r*, equivalently, https://oeis.org/A001917 at the entry of the prime *r* does not divide https://oeis.org/A094593 at the entry of the prime *r*), e.g. the case of *b* = 128 (i.e. *r* = 7) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 131072 (i.e. *r* = 17) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 17 (i.e. the odd primes *p* == 3, 5, 6, 7, 10, 11, 12, 14 mod 17) (i.e. the odd primes *p* in https://oeis.org/A038890); and the case of *b* = 2147483648 (i.e. *r* = 31) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes *p* == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes *p* in https://oeis.org/A191067), also combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not cubic residues (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod 31 (i.e. the odd primes *p* == 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28 mod 31); and the case of *b* = 34359738368 (i.e. *r* = 35) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 except *p* = 5 (i.e. the odd primes *p* == 3, 5, 6 mod 7 except *p* = 5) (i.e. the odd primes *p* in https://oeis.org/A003625 except *p* = 5); and the case of *b* = 2199023255552 (i.e. *r* = 41) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 41 (i.e. the odd primes *p* == 3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38 mod 41) (i.e. the odd primes *p* in https://oeis.org/A038920); and the case of *b* = 8796093022208 (i.e. *r* = 43) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not cubic residues (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod 43 (i.e. the odd primes *p* == 3, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 36, 37, 38, 40 mod 43); and the case of *b* = 562949953421312 (i.e. *r* = 49) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes *p* == 3, 5, 6 mod 7) (i.e. the odd primes *p* in https://oeis.org/A003625); and the case of *b* = 36028797018963968 (i.e. *r* = 55) is combine of sum-of-two-*p*th-powers factorization for the odd primes *p* which are quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 5 or 11 but not both (i.e. the odd primes *p* == 3, 6, 12, 19, 21, 23, 24, 27, 29, 37, 38, 39, 41, 42, 46, 47, 48, 51, 53, 54 mod 55) (i.e. the odd primes *p* in https://oeis.org/A191074); etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, https://t5k.org/notes/Dirichlet.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes))|23 (119216)
53 (227184)
68 (320)|
|8×*b**n*−1|*x*{*y*}|7|*b*−1|9|1||**(such base *b* does not exist if *n* is divisible by 3 because of the difference-of-two-cubes factorization)**|https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C321%7C800000%7C*b* == 1 mod 7: always divisible by 7
*b* == 20 mod 21: always divisible by some element of {3,7}
*b* == 83, 307 mod 455: always divisible by some element of {5,7,13}
*b* = *m*3: difference-of-two-cubes factorization
*b* == 1266, 13593, 27292, 46353 mod 63973: combine of factors {7,13,19,37} and difference-of-two-cubes factorization|97 (192336)
101 (113)
112 (269)
131 (197)
145 (6369)
170 (15423)
194 (38361)
202 (155772)
217 (179)
237 (528)
245 (501)
252 (6288)
270 (108)
277 (1229)
282 (21413)
283 (164769)
284 (5267)|
|9×*b**n*+1|*x*{0}*y*|9|1|10|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C724%7C600000%7C*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 5: always divisible by 5|94 (264)
134 (184)
182 (264)
244 (1836)
248 (39511)
332 (311)
334 (340)
344 (306)
364 (166)
400 (265)
402 (127)
422 (106)
448 (372)
454 (136)
490 (469)
534 (106)
544 (4706)
592 (96870)
622 (127)
634 (190)
664 (290)|
|9×*b**n*−1|*x*{*y*}|8|*b*−1|10|1||**(such base *b* does not exist if *n* is even because of the difference-of-two-squares factorization)**|https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C378%7C300000%7C*b* == 1 mod 2: always divisible by 2
*b* = *m*2: difference-of-two-squares factorization
*b* == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization|88 (172)
112 (5718)
116 (250)
130 (468)
138 (35686)
188 (3888)
198 (304)
218 (178)
258 (106)
286 (164)
292 (2928)
328 (606)
332 (946)
346 (130)
360 (316)
366 (238)|
|10×*b**n*+1|*x*{0}*y*|10|1|11|1|https://oeis.org/A088782
https://oeis.org/A088622 (corresponding primes)|https://oeis.org/A089319
https://oeis.org/A089318 (corresponding primes)|https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C185%7C1000000%7C*b* == 1 mod 11: always divisible by 11
*b* == 32 mod 33: always divisible by some element of {3,11}|17 (1357)
61 (166)
74 (139)
101 (1507)
137 (103)
142 (408)
173 (264235)
176 (147)
179 (337)|
|10×*b**n*−1|*x*{*y*}|9|*b*−1|11|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C233%7C1000000%7C*b* == 1 mod 3: always divisible by 3
*b* == 32 mod 33: always divisible by some element of {3,11}|17 (118)
80 (423716)
89 (250)
185 (6784)
194 (3150)
215 (144)|
|11×*b**n*+1|*x*{0}*y*|11|1|12|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C560%7C100000%7C*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 3: always divisible by 3
*b* == 14 mod 15: always divisible by some element of {3,5}|68 (3948)
108 (190)
110 (162)
152 (838)
222 (101)
236 (154)
294 (365)
320 (1264)
384 (491)
392 (412)
432 (226)
440 (146)
462 (762)
506 (270)
528 (249)
534 (689)
542 (4910)|
|11×*b**n*−1|*x*{*y*}|10|*b*−1|12|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C214%7C1000000%7C*b* == 1 mod 2: always divisible by 2
*b* == 1 mod 5: always divisible by 5
*b* == 14 mod 15: always divisible by some element of {3,5}
*b* = 11×*m*2 with *m* == 2, 3 mod 5: combine of factor 5 and difference-of-two-squares factorization|38 (767)
68 (199)
72 (2446)
80 (209)
102 (2071)
140 (109)
170 (109)
178 (178)
188 (183)|
|12×*b**n*+1|*x*{0}*y*|12|1|13|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C163%7C500000%7C*b* == 1 mod 13: always divisible by 13
*b* == 142 mod 143: always divisible by some element of {11,13}
*b* == 562, 828, 900, 1166 mod 1729: always divisible by some element of {7,13,19}
*b* == 597, 1143 mod 1885: always divisible by some element of {5,13,29}
*b* == 296, 901, 1759, 3090, 4553, 5521, 5807, 6016, 6984, 7094, 7270, 7380, 7479, 8447, 8557, 8733, 8843, 9910, 10020, 10196, 10306, 11483, 11769, 12737, 14200, 15531, 16994, 18457 mod 19019: always divisible by some element of {7,11,13,19}
*b* == 563, 1433, 13212, 15097, 19848, 20718, 32497, 34382, 39133, 51782, 53667, 58418, 58452, 60337, 60883, 71067, 72952, 77737, 79622, 80168, 94267, 97022, 98583, 98907, 113552, 116307, 117868, 118192, 131967, 132513, 132837, 134398, 151252, 151798, 152122, 153683, 170537, 171083, 172968, 177753, 179638, 189822, 190368, 192253, 192287, 197038, 198923, 211572, 213568, 216323, 218208, 229987, 232853, 235608, 237493, 249272 mod 250705: always divisible by some element of {5,7,13,19,29}|30 (1024)
65 (685)
67 (136)
68 (656922)
82 (108)
87 (1215)
102 (2740)
106 (139)
159 (122)|
|12×*b**n*−1|*x*{*y*}|11|*b*−1|13|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354%7C263%7C314000%7C*b* == 1 mod 11: always divisible by 11
*b* == 142 mod 143: always divisible by some element of {11,13}
*b* == 307, 1143 mod 1595: always divisible by some element of {5,11,29}
*b* == 901, 6016, 7479, 18457 mod 19019: always divisible by some element of {7,11,13,19}|43 (204)
65 (1194)
98 (3600)
129 (229)
147 (113)
153 (21660)
186 (112718)
193 (117)
230 (188)|
|(*b*−1)×*b**n*+1|*x*{0}*y*|*b*−1|1|2|1|https://oeis.org/A305531
https://oeis.org/A087139 (prime *b*, *n* replaced by *n*+1)|**(such base *b* does not exist if *n* == 1 mod 6 except *n* = 1 because such numbers are divisible by *b*2−*b*+1)**|https://www.rieselprime.de/ziki/Williams_prime_MP_least
https://www.rieselprime.de/ziki/Williams_prime_MP_table
https://www.rieselprime.de/ziki/Williams_prime_MP_remaining
https://pzktupel.de/Primetables/TableWilliams2.php
https://pzktupel.de/Primetables/Williams_2.txt
https://web.archive.org/web/20240126201446/https://pzktupel.de/Primetables/Williams2DB.txt
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/477 (in Italian)|342|300000|(none)|53 (961)
65 (947)
77 (829)
88 (3023)
122 (6217)
123 (865891)
127 (166)
136 (280)
158 (1621)
180 (2485)
182 (397)
185 (209)
197 (521)
202 (46774)
214 (119)
248 (605)
249 (1852)
251 (102979)
257 (1345)
269 (1437)
272 (16681)
275 (981)
282 (277)
297 (14314)
298 (60671)
307 (204)
317 (129)
319 (565)
326 (64757)
328 (1627)
329 (481)
332 (113)
338 (273)
340 (325)|
|(*b*−1)×*b**n*−1|*x*{*y*}|*b*−2|*b*−1|2|1|https://oeis.org/A122396 (prime *b*, *n* replaced by *n*+1)|**(such base *b* does not exist if *n* == 4 mod 6 because such numbers are divisible by *b*2−*b*+1)**|https://harvey563.tripod.com/wills.txt
https://www.rieselprime.de/ziki/Williams_prime_MM_least
https://www.rieselprime.de/ziki/Williams_prime_MM_table
https://www.rieselprime.de/ziki/Williams_prime_MM_remaining
https://pzktupel.de/Primetables/TableWilliams1.php
https://pzktupel.de/Primetables/Williams_1.txt
https://web.archive.org/web/20240126201427/https://pzktupel.de/Primetables/Williams1DB.txt
https://sites.google.com/view/williams-primes
http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_9.pdf)
https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_10.pdf)
http://www.bitman.name/math/table/484 (in Italian)|128|2450000|(none)|26 (134)
38 (136212)
62 (900)
83 (21496)
91 (520)
93 (477)
98 (4984)
108 (411)
113 (286644)
125 (8740)|
|*b**n*+(*b*−1)|*x*{0}*y*|1|*b*−1|2|1|https://oeis.org/A076845
https://oeis.org/A076846 (corresponding primes)
https://oeis.org/A078178 (*n* = 1 not allowed)
https://oeis.org/A078179 (*n* = 1 not allowed, corresponding primes)|https://oeis.org/A248079
**(such base *b* does not exist if *n* == 5 mod 6 because such numbers are divisible by *b*2−*b*+1)**|https://pzktupel.de/Primetables/TableWilliams6.php
https://pzktupel.de/Primetables/W6DB.txt
https://web.archive.org/web/20231015225001/https://pzktupel.de/Primetables/Williams6DB.txt
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/795 (in Italian)|257|17000|(none)|32 (109)
80 (195)
107 (1401)
113 (20089)
123 (64371)
128 (505)
161 (105)
173 (11429)
179 (3357)
197 (977)
212 (109)
224 (259)
227 (157)
237 (110)
238 (117)|
|*b**n*−(*b*−1)|{*x*}*y*|*b*−1|1|2|2|https://oeis.org/A113516
https://oeis.org/A343589 (corresponding primes)|https://oeis.org/A113517
**(such base *b* does not exist if *n* == 2 mod 6 except *n* = 2 because such numbers are divisible by *b*2−*b*+1)**|https://pzktupel.de/Primetables/TableWilliams5.php
https://pzktupel.de/Primetables/W5DB.txt
https://web.archive.org/web/20231015225036/https://pzktupel.de/Primetables/Williams5DB.txt
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/792 (in Italian)
https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html (prime *b*)
http://www.bitman.name/math/table/435 (in Italian) (prime *b*)|93|60000|(none)|71 (3019)
82 (169)
83 (965)
88 (2848)|
|((*b*−2)×*b**n*+1)/(*b*−1)|{*x*}*y*|*b*−2|*b*−1|3|2|https://oeis.org/A243404 (*n* = 1 allowed)|https://oeis.org/A243341%7Chttps://sites.google.com/view/repunit-and-antirepunit%7C143%7C10000%7C(none)%7C13 (564)
33 (252)
70 (555)
83 (680)
89 (132)
91 (140)
98 (137)
108 (492)
136 (155)|