Editing Quadratic sieve (section)

===Partial relations and cycles===
Even if for some relation ''y''(''x'') is not smooth, it may be possible to merge two of these ''partial relations'' to form a full one, if the two ''y''{{'}}s are products of the same prime(s) outside the factor base. [Note that this is equivalent to extending the factor base.] For example, if the factor base is {2, 3, 5, 7} and ''n'' = 91, there are partial relations:

:<math>{21^2\equiv 7^1\cdot 11\pmod{91}}</math>
:<math>{29^2\equiv 2^1\cdot 11\pmod{91}}</math>

Multiply these together:

:<math>{(21\cdot 29)^2\equiv2^1\cdot7^1\cdot11^2\pmod{91}}</math>

and multiply both sides by (11<sup>&minus;1</sup>)<sup>2</sup> modulo 91. 11<sup>&minus;1</sup> modulo 91 is 58, so:

:<math>(58\cdot 21\cdot 29)^2\equiv 2^1\cdot7^1\pmod{91}</math>
:<math>14^2\equiv 2^1\cdot7^1\pmod{91}</math>

producing a full relation. Such a full relation (obtained by combining partial relations) is called a ''cycle''. Sometimes, forming a cycle from two partial relations leads directly to a congruence of squares, but rarely.