Editing Euler numbers (section)

==Congruences==
W. Zhang<ref>{{cite journal | first1=W.P.| last1= Zhang | title=Some identities involving the Euler and the central factorial numbers | journal=Fibonacci Quarterly | volume=36 | issue=4 | pages=154–157 | year=1998 | doi= 10.1080/00150517.1998.12428950 | url= https://www.mathstat.dal.ca/FQ/Scanned/36-2/zhang.pdf |archive-url=https://web.archive.org/web/20191123004402/https://www.mathstat.dal.ca/FQ/Scanned/36-2/zhang.pdf |archive-date=2019-11-23 |url-status=live}}</ref> obtained the following combinational identities concerning the Euler numbers.  For any prime <math> p </math>, we have
:<math>
(-1)^{\frac{p-1}{2}} E_{p-1} \equiv \textstyle\begin{cases} \phantom{-} 0 \mod p &\text{if }p\equiv 1\bmod 4; \\ -2 \mod p & \text{if }p\equiv 3\bmod 4. \end{cases}
</math>
W. Zhang and Z. Xu<ref>{{cite journal | first1=W.P. | last1= Zhang | first2= Z.F. | last2=Xu | title=On a conjecture of the Euler numbers | journal=Journal of Number Theory | volume=127 | issue=2| pages= 283–291 | year=2007 | doi= 10.1016/j.jnt.2007.04.004 | doi-access=free }}
</ref> proved that, for any prime <math>p \equiv 1 \pmod{4}</math> and integer <math> \alpha\geq 1 </math>, we have
:<math>  E_{\phi(p^{\alpha})/2}\not \equiv 0 \pmod{p^{\alpha}}, </math>
where <math>\phi(n)</math> is the [[Euler's totient function]].