Editing Lucas–Lehmer primality test (section)

===Alternate starting values===

Starting values ''s''<sub>0</sub> other than 4 are possible, for instance 10, 52, and others {{OEIS|id=A018844}}.<ref name="Jansen">{{cite thesis |last=Jansen |first=B.J.H. |title=Mersenne primes and class field theory |pages=iii–iv |date=2012 |type=Doctoral thesis |institution=Leiden University |url=https://openaccess.leidenuniv.nl/handle/1887/20310 |access-date=2018-12-17}}</ref> The Lucas–Lehmer residue calculated with these alternative starting values will still be zero if ''M''<sub>''p''</sub> is a Mersenne prime. However, the terms of the sequence will be different and a non-zero Lucas-Lehmer residue for non-prime ''M''<sub>''p''</sub> will have a different numerical value from the non-zero value calculated when ''s''<sub>0</sub>&nbsp;=&nbsp;4.

It is also possible to use the starting value (2&nbsp;mod&nbsp;''M''<sub>''p''</sub>)(3&nbsp;mod&nbsp;''M''<sub>''p''</sub>)<sup>−1</sup>, usually denoted by 2/3 for short.<ref name="Jansen"/> This starting value equals (2<sup>p</sup>&nbsp;+&nbsp;1)&nbsp;/3, the [[Wagstaff number]] with exponent ''p''.

Starting values like 4, 10, and 2/3 are universal, that is, they are valid for all (or nearly all) ''p''. There are infinitely many additional universal starting values.<ref name="Jansen"/> However, some other starting values are only valid for a subset of all possible ''p'', for example ''s''<sub>0</sub>&nbsp;=&nbsp;3 can be used if ''p''&nbsp;=&nbsp;3&nbsp;(mod&nbsp;4).<ref>{{cite journal |first=Raphael M. |last=Robinson |date=1954 |title=Mersenne and Fermat numbers |journal=Proc. Amer. Math. Soc. |volume=5 |issue=5 |pages=842–846 |doi=10.1090/S0002-9939-1954-0064787-4|doi-access=free }}</ref> This starting value was often used where suitable in the era of hand computation, including by Lucas in proving ''M''<sub>''127''</sub> prime.<ref>{{cite tech report |url=https://core.ac.uk/download/pdf/12758.pdf?repositoryId=17 |first=Guy |last=Haworth |title=Mersenne numbers |page=20 |year=1990 |access-date=2018-12-17}}</ref>
The first few terms of the sequence are 3, 7, 47, ... {{OEIS|A001566}}.