Editing Parity (mathematics) (section)

===Number theory===
The even numbers form an [[ring ideal|ideal]] in the [[ring (algebra)|ring]] of integers,<ref>{{citation|title=Elements of Number Theory|first=John|last=Stillwell|author-link=John Stillwell|publisher=Springer|year=2003 |isbn=9780387955872|page=199 |url=https://books.google.com/books?id=LiAlZO2ntKAC&pg=PA199}}.</ref> but the odd numbers do not&mdash;this is clear from the fact that the [[Identity (mathematics)|identity]] element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 [[modular arithmetic|modulo]] this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.

All [[prime number]]s are odd, with one exception: the prime number 2.<ref>{{citation |last1=Lial |first1=Margaret L. |title=Basic College Mathematics |url=https://archive.org/details/basiccollegemath00lial/page/128/mode/2up |page=128 |year=2005 |edition=7th |publisher=Addison Wesley |isbn=9780321257802 |last2=Salzman |first2=Stanley A. |last3=Hestwood |first3=Diana}}.</ref> All known [[perfect number]]s are even; it is unknown whether any odd perfect numbers exist.<ref>{{citation|title=Mathematical Cranks|title-link=Mathematical Cranks|series=MAA Spectrum| first=Underwood| last=Dudley|author-link=Underwood Dudley|publisher=Cambridge University Press|year=1992|contribution=Perfect numbers| pages=242–244| contribution-url=https://books.google.com/books?id=HqeoWPsIH6EC&pg=PA242|isbn=9780883855072}}.</ref>

[[Goldbach's conjecture]] states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern [[computer]] calculations have shown this conjecture to be true for integers up to at least 4 &times; 10<sup>18</sup>, but still no general [[mathematical proof|proof]] has been found.<ref>{{citation|title=Empirical verification of the even Goldbach conjecture, and computation of prime gaps, up to 4&middot;10<sup>18</sup>|url=https://www.ams.org/editflow/editorial/uploads/mcom/accepted/120521-Silva/120521-Silva-v2.pdf|first1=Tomás|last1=Oliveira e Silva|first2=Siegfried|last2=Herzog|first3=Silvio|last3=Pardi|journal=Mathematics of Computation|volume=83|issue=288|pages=2033–2060|year=2013|doi=10.1090/s0025-5718-2013-02787-1|doi-access=free}}. In press.</ref>