Kunerth's algorithm (determine modular square root of a number)

[hermann@stamm-wilbrandt.de]

In thread "sqrt(x,n) for non-prime n?" ...
https://pari.math.u-bordeaux.fr/archives/pari-users-2311/msg00035.html

Bill showed how PARI/GP issquare() computes modular square root for 
non-prime modulus:

? issquare(Mod(-1,125),&s)
%1 = 1
? s^2
%2 = Mod(124, 125)
?


Kunerth's algorithm is described here:
https://en.wikipedia.org/wiki/Kunerth%27s_algorithm

Aurel stated that being able to compute square root mod n allows to 
factor n.

So Kunerth's algorithm (from 1878) might be wrong.
On the other hand several examples on Wiki page and talk pages worked.


Yesterday I searched for the original two Kunerth articles from 
reference 1 of Wiki page.
I was surprised to find a hit on amazon:
https://www.amazon.com/Kunerths-Modular-Square-Algorithm-Explained-ebook/dp/B09K3TCQR1/ref=sr_1_1

I invested 1$ for 47 (small) pages ebook on Kunerth's algorihm, that can 
be read with browser.
The ebook contains many examples computed with Mathematica.
I use Mathematica sometimes for stuff that I don't know how to do with 
PARI/GP,
like FindInstance for getting solutions to non-trivial equations.
Like the ebook author I do that on Raspberry Pi computer, because 
Mathematica is
free to use there.


Later Bill helped me to get German language pages 327-346 from 1878 in 
this comment:
https://math.stackexchange.com/questions/95443/how-to-compute-modular-square-roots-when-modulus-is-non-prime#comment9832085_252186

I am German, so reading is not a problem (Kunerth was an Austrian 
scientist).
Some funny words are used, mathematical language changed a bit in last 
145 years ;-)


Regards,

Hermann.