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|*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(2^2×*r*+1)|2^2×*r*+1−2^*r*+1+1|2^2×*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(3^2×*r*+1)|3^2×*r*+1−3^*r*+1+1|3^2×*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(5^2×*r*+1)|5^4×*r*+2−5^3×*r*+2+3×5^2×*r*+1−5^*r*+1+1|5^4×*r*+2+5^3×*r*+2+3×5^2×*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(6^2×*r*+1)|6^4×*r*+2−6^3×*r*+2+3×6^2×*r*+1−6^*r*+1+1|6^4×*r*+2+6^3×*r*+2+3×6^2×*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(7^2×*r*+1)|7^6×*r*+3−7^5×*r*+3+3×7^4×*r*+2−7^3×*r*+2+3×7^2×*r*+1−7^*r*+1+1|7^6×*r*+3+7^5×*r*+3+3×7^4×*r*+2+7^3×*r*+2+3×7^2×*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(10^2×*r*+1)|10^8×*r*+4−10^7×*r*+4+5×10^6×*r*+3−2×10^5×*r*+3+7×10^4×*r*+2−2×10^3×*r*+2+5×10^2×*r*+1−10^*r*+1+1|10^8×*r*+4+10^7×*r*+4+5×10^6×*r*+3+2×10^5×*r*+3+7×10^4×*r*+2+2×10^3×*r*+2+5×10^2×*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(11^2×*r*+1)|11^10×*r*+5−11^9×*r*+5+5×11^8×*r*+4−11^7×*r*+4−11^6×*r*+3+11^5×*r*+3−11^4×*r*+2−11^3×*r*+2+5×11^2×*r*+1−11^*r*+1+1|11^10×*r*+5+11^9×*r*+5+5×11^8×*r*+4+11^7×*r*+4−11^6×*r*+3−11^5×*r*+3−11^4×*r*+2+11^3×*r*+2+5×11^2×*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(12^2×*r*+1)|12^2×*r*+1−12^*r*+1/2+1|12^2×*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(13^2×*r*+1)|13^12×*r*+6−13^11×*r*+6+7×13^10×*r*+5−3×13^9×*r*+5+15×13^8×*r*+4−5×13^7×*r*+4+19×13^6×*r*+3−5×13^5×*r*+3+15×13^4×*r*+2−3×13^3×*r*+2+7×13^2×*r*+1−13^*r*+1+1|13^12×*r*+6+13^11×*r*+6+7×13^10×*r*+5+3×13^9×*r*+5+15×13^8×*r*+4+5×13^7×*r*+4+19×13^6×*r*+3+5×13^5×*r*+3+15×13^4×*r*+2+3×13^3×*r*+2+7×13^2×*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(14^2×*r*+1)|14^12×*r*+6−14^11×*r*+6+7×14^10×*r*+5−2×14^9×*r*+5+3×14^8×*r*+4+14^7×*r*+4−7×14^6×*r*+3+14^5×*r*+3+3×14^4×*r*+2−2×14^3×*r*+2+7×14^2×*r*+1−14^*r*+1+1|14^12×*r*+6+14^11×*r*+6+7×14^10×*r*+5+2×14^9×*r*+5+3×14^8×*r*+4-14^7×*r*+4-7×14^6×*r*+3-14^5×*r*+3+3×14^4×*r*+2+2×14^3×*r*+2+7×14^2×*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(15^2×*r*+1)|15^8×*r*+4−15^7×*r*+4+8×15^6×*r*+3−3×15^5×*r*+3+13×15^4×*r*+2−3×15^3×*r*+2+8×15^2×*r*+1−15^*r*+1+1|15^8×*r*+4+15^7×*r*+4+8×15^6×*r*+3+3×15^5×*r*+3+13×15^4×*r*+2+3×15^3×*r*+2+8×15^2×*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

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 *     Feel free to try your editing skills below     *
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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|–|–|