Editing Twin prime (section)

==Other elementary properties==
Every third [[parity (mathematics)|odd]] number is divisible by 3, and therefore no three successive odd numbers can be prime unless one of them is 3. Therefore, 5 is the only prime that is part of two twin prime pairs. The lower member of a pair is by definition a [[Chen prime]].

If ''m''&nbsp;−&nbsp;4 or ''m''&nbsp;+&nbsp;6 is also prime then the three primes are called a [[prime triplet]].

It has been proven<ref>{{cite journal
 |author=P. A. Clement
 |date=January 1949
 |title=Congruences for sets of primes
 |journal=[[American Mathematical Monthly]]
 |volume=56 |issue=1 |pages=23–25
 |doi=10.2307/2305816
 |jstor=2305816
 |url=http://www.math.stonybrook.edu/~moira/mat331-spr10/papers/1949%20ClementCongruences%20for%20Sets%20of%20Primes.pdf
}}</ref> that the pair (''m'',&nbsp;''m''&nbsp;+&nbsp;2) is a twin prime if and only if

:<math>4((m-1)! + 1) \equiv -m \pmod {m(m+2)}.</math>

For a twin prime pair of the form (6''n'' − 1, 6''n'' + 1) for some natural number ''n'' > 1, ''n'' must end in the digit 0, 2, 3, 5, 7, or 8 ({{OEIS2C|id=A002822}}).  If ''n'' were to end in 1 or 6, 6''n'' would end in 6, and 6''n''&thinsp;−1 would be a multiple of 5.  This is not prime unless ''n''&nbsp;=&nbsp;1.  Likewise, if ''n'' were to end in 4 or 9, 6''n'' would end in 4, and 6''n''&thinsp;+1 would be a multiple of 5.  The same rule applies modulo any prime ''p'' ≥ 5: If ''n'' ≡ ±6<sup>−1</sup> (mod ''p''), then one of the pair will be divisible by ''p'' and will not be a twin prime pair unless 6''n'' = ''p''&thinsp;±1.  ''p'' = 5 just happens to produce particularly simple patterns in base 10.