Editing Counterexample (section)

===Other mathematical examples===
{{See also|Counterexamples in topology|Minimal counterexample}}

A counterexample to the statement "all [[prime number]]s are [[Parity (mathematics)|odd numbers]]" is the number 2, as it is a prime number but is not an odd number.<ref name=":0" /> Neither of the numbers 7 or 10 is a counterexample, as neither of them are enough to contradict the statement. In this example, 2 is in fact the only possible counterexample to the statement, even though that alone is enough to contradict the statement. In a similar manner, the statement "All [[natural number]]s are either [[Prime number|prime]] or [[Composite number|composite]]" has the number 1 as a counterexample, as 1 is neither prime nor composite.

[[Euler's sum of powers conjecture]] was disproved by counterexample. It asserted that at least ''n'' ''n''<sup>th</sup> powers were necessary to sum to another ''n''<sup>th</sup> power. This conjecture was disproved in 1966,<ref>{{cite journal|last=Lander, Parkin|year=1966|title=Counterexample to Euler's conjecture on sums of like powers|journal=Bulletin of the American Mathematical Society|publisher=Americal Mathematical Society|volume=72|issue=6|page=1079|issn=0273-0979|url=https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf|access-date=2 August 2018|doi=10.1090/s0002-9904-1966-11654-3|doi-access=free}}</ref> with a counterexample involving ''n''&nbsp;=&nbsp;5; other ''n''&nbsp;=&nbsp;5 counterexamples are now known, as well as some ''n''&nbsp;=&nbsp;4 counterexamples.<ref>{{Cite journal|last=Elkies|first=Noam|date=October 1988|title=On A4 + B4 + C4 = D4|url=https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-1988-0930224-9/S0025-5718-1988-0930224-9.pdf|journal=Mathematics of Computation|volume=51|issue=184|pages=825–835}}</ref>

[[Witsenhausen's counterexample]] shows that it is not always true (for [[control theory|control problems]]) that a quadratic [[loss function]] and a linear equation of evolution of the [[state variable]] imply optimal control laws that are linear.

All [[Euclidean plane isometries]] are mappings that preserve [[area]], but the [[converse (logic)|converse]] is false as shown by counterexamples [[shear mapping]] and [[squeeze mapping]].

Other examples include the disproofs of the [[Seifert conjecture]], the [[Pólya conjecture]], the conjecture of [[Hilbert's fourteenth problem]], [[Tait's conjecture]], and the [[Ganea conjecture]].