;Task.

Convert rational a/b to its approximate [[wp:Hensel%27s_lemma#Hensel's_lemma_for_p-adic_numbers|p-adic square root]]. To check the result,
square the root and construct rational m/n to compare with radicand a/b.

For rational reconstruction Lagrange's [[wp:Lattice_reduction|lattice basis reduction]] algorithm is used.

'''Recipe:''' find root {{math|''x<sub>1</sub>'' modulo p}} and build a sequence of solutions
{{math|''f''(''x<sub>k</sub>'') &#8801; 0 (mod p<sup>k</sup>)}},
<br/>using the lifting equation
{{math|''x<sub>k+1</sub>'' &#61; ''x<sub>k</sub>'' + ''d<sub>k</sub>'' * ''p<sup>k</sup>''&nbsp;}} 
with {{math|''d<sub>k</sub>'' &#61; &#8211;(''f''(''x<sub>k</sub>'') &#47; ''p<sup>k</sup>'') &#47;
''f''&nbsp;&#8242;(''x<sub>1</sub>'') (mod p)}}.
<br/>The multipliers {{math|''d<sub>k</sub>''}} are the successive p-adic digits to find.

If evaluation of {{math|''f''(''x'') &#61; ''bx<sup>2</sup>'' &#8211; ''a''}} overflows,
the expansion is cut off and might be too short to retrieve the radicand.
Setting a higher precision won't help, using a programming language with built-in
large integer support will.


;Related task.

[[p-Adic numbers, basic]]


;Reference.

[https://www.uvm.edu/~cvincen1/files/teaching/spring2017-math255/quadraticequation.pdf]
Solving {{math|''x<sup>2</sup>'' &#8801; ''a'' (mod n)}}


__TOC__