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|*b*|the algebraic factor of *b*<sup>*n*</sup>±1 which has Aurifeuillean factorization|the Aurifeuillean *L* factor|the Aurifeuillean *M* factor|examples of the Aurifeuillean factorization for exponent 2×*r*+1 = 37 (see the "Aurifeuillian" section)<br>(note that 37 is prime and > 36)|*OEIS* sequence of the Aurifeuillean *L* factor|*OEIS* sequence of the Aurifeuillean *M* factor|
|---|---|---|---|---|---|---|
|2, 4, 8, 16, 32|*Φ*<sub>4</sub>(2<sup>2×*r*+1</sup>)|2<sup>2×*r*+1</sup>−2<sup>*r*+1</sup>+1|2<sup>2×*r*+1</sup>+2<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=2&Exp=74&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A092440|https://oeis.org/A085601|
|3, 9, 27|*Φ*<sub>6</sub>(3<sup>2×*r*+1</sup>)|3<sup>2×*r*+1</sup>−3<sup>*r*+1</sup>+1|3<sup>2×*r*+1</sup>+3<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=3&Exp=111&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220978|https://oeis.org/A198410|
|5, 25|*Φ*<sub>5</sub>(5<sup>2×*r*+1</sup>)|5<sup>4×*r*+2</sup>−5<sup>3×*r*+2</sup>+3×5<sup>2×*r*+1</sup>−5<sup>*r*+1</sup>+1|5<sup>4×*r*+2</sup>+5<sup>3×*r*+2</sup>+3×5<sup>2×*r*+1</sup>+5<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=5&Exp=185&LBIDPMList=A&LBIDLODList=D|https://oeis.org/A220979|https://oeis.org/A220980|
|6, 36|*Φ*<sub>12</sub>(6<sup>2×*r*+1</sup>)|6<sup>4×*r*+2</sup>−6<sup>3×*r*+2</sup>+3×6<sup>2×*r*+1</sup>−6<sup>*r*+1</sup>+1|6<sup>4×*r*+2</sup>+6<sup>3×*r*+2</sup>+3×6<sup>2×*r*+1</sup>+6<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=6&Exp=222&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220981|https://oeis.org/A220982|
|7|*Φ*<sub>14</sub>(7<sup>2×*r*+1</sup>)|7<sup>6×*r*+3</sup>−7<sup>5×*r*+3</sup>+3×7<sup>4×*r*+2</sup>−7<sup>3×*r*+2</sup>+3×7<sup>2×*r*+1</sup>−7<sup>*r*+1</sup>+1|7<sup>6×*r*+3</sup>+7<sup>5×*r*+3</sup>+3×7<sup>4×*r*+2</sup>+7<sup>3×*r*+2</sup>+3×7<sup>2×*r*+1</sup>+7<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=7&Exp=259&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220983|https://oeis.org/A220984|
|10|*Φ*<sub>20</sub>(10<sup>2×*r*+1</sup>)|10<sup>8×*r*+4</sup>−10<sup>7×*r*+4</sup>+5×10<sup>6×*r*+3</sup>−2×10<sup>5×*r*+3</sup>+7×10<sup>4×*r*+2</sup>−2×10<sup>3×*r*+2</sup>+5×10<sup>2×*r*+1</sup>−10<sup>*r*+1</sup>+1|10<sup>8×*r*+4</sup>+10<sup>7×*r*+4</sup>+5×10<sup>6×*r*+3</sup>+2×10<sup>5×*r*+3</sup>+7×10<sup>4×*r*+2</sup>+2×10<sup>3×*r*+2</sup>+5×10<sup>2×*r*+1</sup>+10<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=10&Exp=370&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220985|https://oeis.org/A220986|
|11|*Φ*<sub>22</sub>(11<sup>2×*r*+1</sup>)|11<sup>10×*r*+5</sup>−11<sup>9×*r*+5</sup>+5×11<sup>8×*r*+4</sup>−11<sup>7×*r*+4</sup>−11<sup>6×*r*+3</sup>+11<sup>5×*r*+3</sup>−11<sup>4×*r*+2</sup>−11<sup>3×*r*+2</sup>+5×11<sup>2×*r*+1</sup>−11<sup>*r*+1</sup>+1|11<sup>10×*r*+5</sup>+11<sup>9×*r*+5</sup>+5×11<sup>8×*r*+4</sup>+11<sup>7×*r*+4</sup>−11<sup>6×*r*+3</sup>−11<sup>5×*r*+3</sup>−11<sup>4×*r*+2</sup>+11<sup>3×*r*+2</sup>+5×11<sup>2×*r*+1</sup>+11<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=11&Exp=407&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220987|https://oeis.org/A220988|
|12|*Φ*<sub>6</sub>(12<sup>2×*r*+1</sup>)|12<sup>2×*r*+1</sup>−12<sup>*r*+1</sup>/2+1|12<sup>2×*r*+1</sup>+12<sup>*r*+1</sup>/2+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=12&Exp=111&LBIDPMList=B&LBIDLODList=D|https://oeis.org/A220989|https://oeis.org/A220990|
|13|*Φ*<sub>13</sub>(13<sup>2×*r*+1</sup>)|13<sup>12×*r*+6</sup>−13<sup>11×*r*+6</sup>+7×13<sup>10×*r*+5</sup>−3×13<sup>9×*r*+5</sup>+15×13<sup>8×*r*+4</sup>−5×13<sup>7×*r*+4</sup>+19×13<sup>6×*r*+3</sup>−5×13<sup>5×*r*+3</sup>+15×13<sup>4×*r*+2</sup>−3×13<sup>3×*r*+2</sup>+7×13<sup>2×*r*+1</sup>−13<sup>*r*+1</sup>+1|13<sup>12×*r*+6</sup>+13<sup>11×*r*+6</sup>+7×13<sup>10×*r*+5</sup>+3×13<sup>9×*r*+5</sup>+15×13<sup>8×*r*+4</sup>+5×13<sup>7×*r*+4</sup>+19×13<sup>6×*r*+3</sup>+5×13<sup>5×*r*+3</sup>+15×13<sup>4×*r*+2</sup>+3×13<sup>3×*r*+2</sup>+7×13<sup>2×*r*+1</sup>+13<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=481&LBIDPMList=A&LBIDLODList=D|–|–|
|14|*Φ*<sub>28</sub>(14<sup>2×*r*+1</sup>)|14<sup>12×*r*+6</sup>−14<sup>11×*r*+6</sup>+7×14<sup>10×*r*+5</sup>−2×14<sup>9×*r*+5</sup>+3×14<sup>8×*r*+4</sup>+14<sup>7×*r*+4</sup>−7×14<sup>6×*r*+3</sup>+14<sup>5×*r*+3</sup>+3×14<sup>4×*r*+2</sup>−2×14<sup>3×*r*+2</sup>+7×14<sup>2×*r*+1</sup>−14<sup>*r*+1</sup>+1|14<sup>12×*r*+6</sup>+14<sup>11×*r*+6</sup>+7×14<sup>10×*r*+5</sup>+2×14<sup>9×*r*+5</sup>+3×14<sup>8×*r*+4</sup>-14<sup>7×*r*+4</sup>-7×14<sup>6×*r*+3</sup>-14<sup>5×*r*+3</sup>+3×14<sup>4×*r*+2</sup>+2×14<sup>3×*r*+2</sup>+7×14<sup>2×*r*+1</sup>+14<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=14&Exp=518&LBIDPMList=B&LBIDLODList=D|–|–|
|15|*Φ*<sub>30</sub>(15<sup>2×*r*+1</sup>)|15<sup>8×*r*+4</sup>−15<sup>7×*r*+4</sup>+8×15<sup>6×*r*+3</sup>−3×15<sup>5×*r*+3</sup>+13×15<sup>4×*r*+2</sup>−3×15<sup>3×*r*+2</sup>+8×15<sup>2×*r*+1</sup>−15<sup>*r*+1</sup>+1|15<sup>8×*r*+4</sup>+15<sup>7×*r*+4</sup>+8×15<sup>6×*r*+3</sup>+3×15<sup>5×*r*+3</sup>+13×15<sup>4×*r*+2</sup>+3×15<sup>3×*r*+2</sup>+8×15<sup>2×*r*+1</sup>+15<sup>*r*+1</sup>+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=15&Exp=555&LBIDPMList=B&LBIDLODList=D|–|–|

Revision as of 14:24, 21 June 2025

|*b*|the algebraic factor of *b**n*±1 which has Aurifeuillean factorization|the Aurifeuillean *L* factor|the Aurifeuillean *M* factor|examples of the Aurifeuillean factorization for exponent 2×*r*+1 = 37 (see the "Aurifeuillian" section)
(note that 37 is prime and > 36)|*OEIS* sequence of the Aurifeuillean *L* factor|*OEIS* sequence of the Aurifeuillean *M* factor| |---|---|---|---|---|---|---| |2, 4, 8, 16, 32|*Φ*4(22×*r*+1)|22×*r*+1−2*r*+1+1|22×*r*+1+2*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=2&Exp=74&LBIDPMList=B&LBIDLODList=D%7Chttps://oeis.org/A092440%7Chttps://oeis.org/A085601%7C |3, 9, 27|*Φ*6(32×*r*+1)|32×*r*+1−3*r*+1+1|32×*r*+1+3*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=3&Exp=111&LBIDPMList=B&LBIDLODList=D%7Chttps://oeis.org/A220978%7Chttps://oeis.org/A198410%7C |5, 25|*Φ*5(52×*r*+1)|54×*r*+2−53×*r*+2+3×52×*r*+1−5*r*+1+1|54×*r*+2+53×*r*+2+3×52×*r*+1+5*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=5&Exp=185&LBIDPMList=A&LBIDLODList=D%7Chttps://oeis.org/A220979%7Chttps://oeis.org/A220980%7C |6, 36|*Φ*12(62×*r*+1)|64×*r*+2−63×*r*+2+3×62×*r*+1−6*r*+1+1|64×*r*+2+63×*r*+2+3×62×*r*+1+6*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=6&Exp=222&LBIDPMList=B&LBIDLODList=D%7Chttps://oeis.org/A220981%7Chttps://oeis.org/A220982%7C |7|*Φ*14(72×*r*+1)|76×*r*+3−75×*r*+3+3×74×*r*+2−73×*r*+2+3×72×*r*+1−7*r*+1+1|76×*r*+3+75×*r*+3+3×74×*r*+2+73×*r*+2+3×72×*r*+1+7*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=7&Exp=259&LBIDPMList=B&LBIDLODList=D%7Chttps://oeis.org/A220983%7Chttps://oeis.org/A220984%7C |10|*Φ*20(102×*r*+1)|108×*r*+4−107×*r*+4+5×106×*r*+3−2×105×*r*+3+7×104×*r*+2−2×103×*r*+2+5×102×*r*+1−10*r*+1+1|108×*r*+4+107×*r*+4+5×106×*r*+3+2×105×*r*+3+7×104×*r*+2+2×103×*r*+2+5×102×*r*+1+10*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=10&Exp=370&LBIDPMList=B&LBIDLODList=D%7Chttps://oeis.org/A220985%7Chttps://oeis.org/A220986%7C |11|*Φ*22(112×*r*+1)|1110×*r*+5−119×*r*+5+5×118×*r*+4−117×*r*+4−116×*r*+3+115×*r*+3−114×*r*+2−113×*r*+2+5×112×*r*+1−11*r*+1+1|1110×*r*+5+119×*r*+5+5×118×*r*+4+117×*r*+4−116×*r*+3−115×*r*+3−114×*r*+2+113×*r*+2+5×112×*r*+1+11*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=11&Exp=407&LBIDPMList=B&LBIDLODList=D%7Chttps://oeis.org/A220987%7Chttps://oeis.org/A220988%7C |12|*Φ*6(122×*r*+1)|122×*r*+1−12*r*+1/2+1|122×*r*+1+12*r*+1/2+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=12&Exp=111&LBIDPMList=B&LBIDLODList=D%7Chttps://oeis.org/A220989%7Chttps://oeis.org/A220990%7C |13|*Φ*13(132×*r*+1)|1312×*r*+6−1311×*r*+6+7×1310×*r*+5−3×139×*r*+5+15×138×*r*+4−5×137×*r*+4+19×136×*r*+3−5×135×*r*+3+15×134×*r*+2−3×133×*r*+2+7×132×*r*+1−13*r*+1+1|1312×*r*+6+1311×*r*+6+7×1310×*r*+5+3×139×*r*+5+15×138×*r*+4+5×137×*r*+4+19×136×*r*+3+5×135×*r*+3+15×134×*r*+2+3×133×*r*+2+7×132×*r*+1+13*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=481&LBIDPMList=A&LBIDLODList=D%7C–%7C–%7C |14|*Φ*28(142×*r*+1)|1412×*r*+6−1411×*r*+6+7×1410×*r*+5−2×149×*r*+5+3×148×*r*+4+147×*r*+4−7×146×*r*+3+145×*r*+3+3×144×*r*+2−2×143×*r*+2+7×142×*r*+1−14*r*+1+1|1412×*r*+6+1411×*r*+6+7×1410×*r*+5+2×149×*r*+5+3×148×*r*+4-147×*r*+4-7×146×*r*+3-145×*r*+3+3×144×*r*+2+2×143×*r*+2+7×142×*r*+1+14*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=14&Exp=518&LBIDPMList=B&LBIDLODList=D%7C–%7C–%7C |15|*Φ*30(152×*r*+1)|158×*r*+4−157×*r*+4+8×156×*r*+3−3×155×*r*+3+13×154×*r*+2−3×153×*r*+2+8×152×*r*+1−15*r*+1+1|158×*r*+4+157×*r*+4+8×156×*r*+3+3×155×*r*+3+13×154×*r*+2+3×153×*r*+2+8×152×*r*+1+15*r*+1+1|http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=15&Exp=555&LBIDPMList=B&LBIDLODList=D%7C–%7C–%7C