This is as yet the last post in the LQR series. If you are only interested in -solving- ( quadratic ) congruence equations then this is the way to do it in Mathematica:
As a general Diophantic equation:
In[1]:= Reduce[x^2==123456+1299709 k,{x,k},Integers]
Out[1]= (C[1]\[Element]Integers&&x==427784-1299709 C[1]&&k==140800-855568 C[1]+1299709 C[1]^2)||(C[1]\[Element]Integers&&x==871925-1299709 C[1]&&k==584941-1743850 C[1]+1299709 C[1]^2)
Or slightly more elegant as a pure congruence equation ( thanks to:
Mr. Wizard ):
In[2]:= Reduce[x^2 == 123456, x, Modulus -> 1299709]
Out[2]= x == 427784 || x == 871925
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