Is the following analogue of Fermat's Little Theorem for Bernoulli numbers true?



> 
> Let $D\_{2n}$ be the denominator of $\frac{B\_{2n}}{4n}$ where $B\_n$ is
> the $n$-th [Bernoulli number](http://oeis.org/A006863). If $\gcd(a, D\_{2n}) = 1$ then
> 
> 
> $$ a^{2n} \equiv 1\pmod{D\_{2n}}.$$
> 
> 
> 


This question was [posted in MSE](https://math.stackexchange.com/questions/4343529/conjectured-analogue-of-fermats-little-theorem-for-bernouli-numbers) 3 weeks back but it is still open. Hence posting in MO.



Analogue of Fermat's little theorem for Bernoulli numbers
A proof is essentially given in Section 5.1 of [Notes on primitive lambda-roots](http://www.maths.qmul.ac.uk/%7Epjc/csgnotes/lambda.pdf) by P. J. Cameron and D. A. Preece.


