Editing Cunningham Project (section)

===Aurifeuillean factors===
{{main|Aurifeuillean factorization}}
When the number is of a particular form (the exact expression varies with the base), aurifeuillean factorization may be used, which gives a product of two or three numbers. The following equations give aurifeuillean factors for the Cunningham project bases as a product of ''F'', ''L'' and ''M'':<ref>{{cite web|title=Main Cunningham Tables|url=https://homes.cerias.purdue.edu/~ssw/cun/pmain125.txt|accessdate=15 January 2025}} At the end of tables 2LM, 3+, 5-, 6+, 7+, 10+, 11+ and 12+ there are formulae detailing the aurifeuillean factorizations.</ref>

Let ''b'' = ''s''<sup>2</sup>{{times}}''k'' with [[squarefree]] ''k'', if one of the conditions holds, then <math>\Phi_n(b)</math> have aurifeuillean factorization.
: (i) <math>k\equiv 1 \pmod 4</math> and <math>n\equiv k \pmod{2k};</math>
: (ii) <math>k\equiv 2, 3 \pmod 4</math> and <math>n\equiv 2k \pmod{4k}.</math>
{{table alignment}}
{| class="wikitable col2right col3right col4center col5center"
!''b''
!Number
!''F''
!''L''
!''M''
!Other definitions
|-
!2
|2<sup>4''k''+2</sup> + 1
|1
|2<sup>2{{itco|''k''}}+1</sup> − 2<sup>{{itco|''k''}}+1</sup> + 1
|2<sup>2{{itco|''k''}}+1</sup> + 2<sup>{{itco|''k''}}+1</sup> + 1
|
|-
!3
|3<sup>6''k''+3</sup> + 1
|3<sup>2{{itco|''k''}}+1</sup> + 1
|3<sup>2{{itco|''k''}}+1</sup> − 3<sup>{{itco|''k''}}+1</sup> + 1
|3<sup>2{{itco|''k''}}+1</sup> + 3<sup>{{itco|''k''}}+1</sup> + 1
|
|-
!5
|5<sup>10''k''+5</sup> − 1
|5<sup>2{{itco|''k''}}+1</sup> − 1
|{{itco|''T''}}<sup>2</sup> − 5<sup>{{itco|''k''}}+1</sup>''T'' + 5<sup>2{{itco|''k''}}+1</sup>
|{{itco|''T''}}<sup>2</sup> + 5<sup>{{itco|''k''}}+1</sup>''T'' + 5<sup>2{{itco|''k''}}+1</sup>
|''T'' = 5<sup>2{{itco|''k''}}+1</sup> + 1
|-
!6
|6<sup>12''k''+6</sup> + 1
|6<sup>4''k''+2</sup> + 1
|{{itco|''T''}}<sup>2</sup> − 6<sup>{{itco|''k''}}+1</sup>''T'' + 6<sup>2{{itco|''k''}}+1</sup>
|{{itco|''T''}}<sup>2</sup> + 6<sup>{{itco|''k''}}+1</sup>''T'' + 6<sup>2{{itco|''k''}}+1</sup>
|''T'' = 6<sup>2{{itco|''k''}}+1</sup> + 1
|-
!7
|7<sup>14''k''+7</sup> + 1
|7<sup>2{{itco|''k''}}+1</sup> + 1
|''A'' − ''B''
|''A'' + ''B''
|''A'' = 7<sup>6''k''+3</sup> + 3(7<sup>4''k''+2</sup>) + 3(7<sup>2{{itco|''k''}}+1</sup>) + 1<br/>''B'' = 7<sup>5''k''+3</sup> + 7<sup>3''k''+2</sup> + 7<sup>{{itco|''k''}}+1</sup>
|-
!10
|10<sup>20{{itco|''k''}}+10</sup> + 1
|10<sup>4''k''+2</sup> + 1
|''A'' − ''B''
|''A'' + ''B''
|''A'' = 10<sup>8''k''+4</sup> + 5(10<sup>6''k''+3</sup>) + 7(10<sup>4''k''+2</sup>) + 5(10<sup>2{{itco|''k''}}+1</sup>) + 1<br/>''B'' = 10<sup>7''k''+4</sup> + 2(10<sup>5''k''+3</sup>) + 2(10<sup>3''k''+2</sup>) + 10<sup>{{itco|''k''}}+1</sup>
|-
!11
|11<sup>22{{itco|''k''}}+11</sup> + 1
|11<sup>2{{itco|''k''}}+1</sup> + 1
|''A'' − ''B''
|''A'' + ''B''
|''A'' = 11<sup>10''k''+5</sup> + 5(11<sup>8''k''+4</sup>) − 11<sup>6''k''+3</sup> − 11<sup>4''k''+2</sup> + 5(11<sup>2{{itco|''k''}}+1</sup>) + 1<br/>''B'' = 11<sup>9''k''+5</sup> + 11<sup>7''k''+4</sup> − 11<sup>5''k''+3</sup> + 11<sup>3''k''+2</sup> + 11<sup>{{itco|''k''}}+1</sup>
|-
!12
|12<sup>6''k''+3</sup> + 1
|12<sup>2{{itco|''k''}}+1</sup> + 1
|12<sup>2{{itco|''k''}}+1</sup> − 6(12<sup>''k''</sup>) + 1
|12<sup>2{{itco|''k''}}+1</sup> + 6(12<sup>''k''</sup>) + 1
|
|}