Editing Strong pseudoprime (section)

==Examples==
The first strong pseudoprimes to base 2 are
:2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, 52633, 65281, 74665, 80581, 85489, 88357, 90751, ... {{OEIS|id=A001262}}.
The first to base 3 are
:121, 703, 1891, 3281, 8401, 8911, 10585, 12403, 16531, 18721, 19345, 23521, 31621, 44287, 47197, 55969, 63139, 74593, 79003, 82513, 87913, 88573, 97567, ... {{OEIS|id=A020229}}.
The first to base 5 are
:781, 1541, 5461, 5611, 7813, 13021, 14981, 15751, 24211, 25351, 29539, 38081, 40501, 44801, 53971, 79381, ... {{OEIS|id=A020231}}.

For base 4, see {{oeis|id=A020230}}, and for base 6 to 100, see {{oeis|id=A020232}} to {{oeis|id=A020326}}.
By testing the above conditions to several bases, one gets somewhat more powerful primality tests than by using one base alone.
For example, there are only 13 numbers less than 25·10<sup>9</sup> that are strong pseudoprimes to bases 2, 3, and 5 simultaneously.
They are listed in Table 7 of.<ref name="PSW"/> The smallest such number is 25326001.
This means that, if ''n'' is less than 25326001 and ''n'' is a strong probable prime to bases 2, 3, and 5, then ''n'' is prime.

Carrying this further, 3825123056546413051 is the smallest number that is a strong pseudoprime to the 9 bases 2, 3, 5, 7, 11, 13, 17, 19, and 23.<ref name="spspii">
{{cite journal|title=Finding Strong Pseudoprimes to Several Bases. II
|journal=Mathematics of Computation|year=2003|volume=72|issue=244|pages=2085–2097
|author1=Zhenxiang Zhang|author2=Min Tang|doi=10.1090/S0025-5718-03-01545-X |doi-access=free}}</ref><ref name="psp9">
{{cite arXiv |last1=Jiang |first1=Yupeng |last2=Deng |first2=Yingpu |eprint=1207.0063v1
|title=Strong pseudoprimes to the first 9 prime bases |class=math.NT |year=2012 }}</ref>
So, if ''n'' is less than 3825123056546413051 and ''n'' is a strong probable prime to these 9 bases, then ''n'' is prime.

By judicious choice of bases that are not necessarily prime, even better tests can be constructed. For example, there is no composite <math>< 2^{64}</math> that is a strong pseudoprime to all of the seven bases 2, 325, 9375, 28178, 450775, 9780504,  and 1795265022.<ref>{{cite web | url=https://miller-rabin.appspot.com | title=SPRP Records | access-date=3 June 2015}}</ref>