Editing Fermat pseudoprime (section)

===Distribution===
There are infinitely many pseudoprimes to any given base <math>a > 1</math>. In 1904, Cipolla showed how to produce an infinite number of pseudoprimes to base <math>a > 1</math>: let <math>A = (a^p-1)/(a-1)</math> and let <math>B = (a^p+1)/(a+1)</math>, where <math>p</math> is a prime number that does not divide <math>a(a^2-1)</math>. Then <math>n= AB</math> is composite, and is a pseudoprime to base <math>a</math>.<ref>{{ cite book | page=108 | author=Paulo Ribenboim |author-link=Paulo Ribenboim | title=The New Book of Prime Number Records | isbn=0-387-94457-5 | publisher=[[Springer-Verlag]] | location=New York | year=1996 }}</ref><ref>{{cite journal | url=https://cs.uwaterloo.ca/journals/JIS/VOL10/Hamahata2/hamahata44.pdf | title=Cipolla Pseudoprimes | last1=Hamahata | first1=Yoshinori | last2=Kokubun | first2=Y. | journal=Journal of Integer Sequences | date=2007 | volume=10| issue=8}}</ref> For example, if <math>a=2</math> and <math>p=5</math>, then <math>A=31</math>, <math>B=11</math>, and <math>n=AB=341</math> is a pseudoprime to base <math>2</math>.

In fact, there are infinitely many [[strong pseudoprime]]s to any base greater than 1 (see Theorem 1 of
<ref name="PSW">{{cite journal |last1=Pomerance |first1=Carl |author-link1=Carl Pomerance |last2=Selfridge |first2=John L. |author-link2=John L. Selfridge |last3=Wagstaff |first3=Samuel S. Jr. |author-link3=Samuel S. Wagstaff Jr. |date=July 1980 |title=The pseudoprimes to 25·10<sup>9</sup> |journal=Mathematics of Computation |doi=10.1090/S0025-5718-1980-0572872-7 |volume=35 |issue=151 |pages=1003–1026 |url=http://www.math.dartmouth.edu/~carlp/PDF/paper25.pdf |archive-url=https://web.archive.org/web/20050304202721/http://math.dartmouth.edu/~carlp/PDF/paper25.pdf |archive-date=2005-03-04 |url-status=live |doi-access=free }}</ref>) and infinitely many Carmichael numbers,<ref name="Alford1994">{{cite journal |last1=Alford |first1=W. R. |author-link=W. R. (Red) Alford |last2=Granville |first2=Andrew |author-link2=Andrew Granville |last3=Pomerance |first3=Carl |author-link3=Carl Pomerance |year=1994 |title=There are Infinitely Many Carmichael Numbers |journal=[[Annals of Mathematics]] |doi=10.2307/2118576 |volume=140 |issue=3 |pages=703–722 |url=http://www.math.dartmouth.edu/~carlp/PDF/paper95.pdf |archive-url=https://web.archive.org/web/20050304203448/http://math.dartmouth.edu/~carlp/PDF/paper95.pdf |archive-date=2005-03-04 |url-status=live |jstor=2118576 }}</ref> but they are comparatively rare. There are three pseudoprimes to base 2 below 1000, 245 below one million, and 21853 less than 25·10<sup>9</sup>. There are 4842 strong pseudoprimes base 2 and 2163 Carmichael numbers below this limit (see Table 1 of <ref name="PSW"/>).

Starting at 17·257, the product of consecutive Fermat numbers is a base-2 pseudoprime, and so are all [[Fermat prime|Fermat composites]] and [[Mersenne prime|Mersenne composites]].

The probability of a composite number n passing the Fermat test approaches zero for <math>n \to\infty</math>. Specifically, Kim and Pomerance showed the following: The probability that a random odd number  <math>n \le x</math> is a Fermat pseudoprime to a random base <math>1<b<n-1</math> is less than 2.77·10<sup>−8</sup> for x= 10<sup>100</sup>, and is at most (log x)<sup>−197</sup><10<sup>−10,000</sup> for x≥10<sup>100,000</sup>.<ref>{{cite journal | url=https://www.jstor.org/stable/2008733 | jstor=2008733 | title=The Probability that a Random Probable Prime is Composite | last1=Kim | first1=Su Hee | last2=Pomerance | first2=Carl | journal=Mathematics of Computation | date=1989 | volume=53 | issue=188 | pages=721–741 | doi=10.2307/2008733 }}</ref>