Editing Pronic number (section)

== Additional properties==
Pronic numbers are even, and 2 is the only [[prime number|prime]] pronic number. It is also the only pronic number in the [[Fibonacci number|Fibonacci sequence]] and the only pronic [[Lucas number]].<ref>{{citation
 | last = McDaniel
 | first = Wayne L.
 | issue = 1
 | journal = [[Fibonacci Quarterly]]
 | mr = 1605345
 | pages = 60–62
 | title = Pronic Lucas numbers
 | url = http://www.mathstat.dal.ca/FQ/Scanned/36-1/mcdaniel2.pdf
 | volume = 36
 | year = 1998
 | doi = 10.1080/00150517.1998.12428962
 | access-date = 2011-05-21
 | archive-url = https://web.archive.org/web/20170705130526/http://www.mathstat.dal.ca/FQ/Scanned/36-1/mcdaniel2.pdf
 | archive-date = 2017-07-05
 | url-status = dead
 }}.</ref><ref>{{citation
 | last = McDaniel | first = Wayne L.
 | issue = 1
 | journal = [[Fibonacci Quarterly]]
 | mr = 1605341
 | pages = 56–59
 | title = Pronic Fibonacci numbers
 | url = http://www.fq.math.ca/Scanned/36-1/mcdaniel1.pdf
 | volume = 36
 | year = 1998| doi = 10.1080/00150517.1998.12428961
 }}.</ref>

The [[arithmetic mean]] of two consecutive pronic numbers is a [[square number]]:
:<math>\frac {n(n+1) + (n+1)(n+2)}{2} = (n+1)^2</math>
So there is a square between any two consecutive pronic numbers. It is unique, since
:<math>n^2 \leq n(n+1) < (n+1)^2 < (n+1)(n+2) < (n+2)^2.</math>
Another consequence of this chain of inequalities is the following property. If {{mvar|m}} is a pronic number, then the following holds:
:<math> \lfloor{\sqrt{m}}\rfloor \cdot \lceil{\sqrt{m}}\rceil = m.</math>

The fact that consecutive integers are [[coprime]] and that a pronic number is the product of two consecutive integers leads to a number of properties.  Each distinct prime factor of a pronic number is present in only one of the factors {{mvar|n}} or {{math|''n'' + 1}}.  Thus a pronic number is [[Square-free integer|squarefree]] if and only if {{mvar|n}} and {{math|''n'' + 1}} are also squarefree.  The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of {{mvar|n}} and {{math|''n'' + 1}}.

If 25 is appended to the [[decimal representation]] of any pronic number, the result is a square number, the square of a number ending on 5; for example, 625 = 25<sup>2</sup> and 1225 = 35<sup>2</sup>. This is so because

:<math>100n(n+1) + 25 = 100n^2 + 100n + 25 = (10n+5)^2</math>.

The difference between two consecutive [[Unit fraction|unit fractions]] is the reciprocal of a pronic number:<ref name="Meyer2024">{{cite web | last=Meyer | first=David | title=A Useful Mathematical Trick, Telescoping Series, and the Infinite Sum of the Reciprocals of the Triangular Numbers | url=https://davidmeyer.github.io/qc/tricks.pdf | page=1 | website=David Meyer's GitHub | access-date=2024-11-26 }}</ref>

:<math>\frac{1}{n} - \frac{1}{n+1} = \frac{(n+1) - n}{n(n+1)} = \frac{1}{n(n+1)}</math>