[was@laptop mlogs]$ [was@laptop mlogs]$ Magma V2.9-11     Fri Oct  4 2002 10:45:52 on laptop   [Seed = 1724314285]
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Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> R := Integers(389);
> R;
Residue class ring of integers modulo 389
> a := R!69;
> a^((389-1)/2);

>> a^((389-1)/2);
    ^
Runtime error in '^': Bad argument types
Argument types given: RngIntResElt, FldRatElt
> n := (389-1)/2;
> n;
194
> Parent(n);
Rational Field
> 4/3;
4/3
> 4 div 3;
1
> a^((389-1) div 2);
1
> n;
194
> n := Integers()!n;
> a^n;
1
> R;
Residue class ring of integers modulo 389
> 
> 
> Z389 := Integers(389);
> S<x> := PolynomialRing(Z389);
> S;
Univariate Polynomial Ring in x over IntegerRing(389)
> R<x> := quo<S | x^2-69>;
> R;
Univariate Quotient Polynomial Algebra in x over Residue class ring of integers modulo 389
with modulus x^2 + 320
> Random(389);
255
> z := 255;
> (1+z*x)^((389-1) div 2);
150*x
> u := Z389!0; v := Z389!150;
> -u/v;
0
> (1-u)/v;
236
> ((1-u)/v)^2;
69
> (-1-u)/v;
153
> ((-1-u)/v)^2;
69
> for z in [1..388] do (1+z*x)^((389-1) div 2); end for;
388
388
150*x
388
388
239*x
1
239*x
1
239*x
388
388
239*x
150*x
388
150*x
239*x
150*x
150*x
388
1
388
1
388
1
150*x
388
1
1
1
1
388
1
150*x
388
239*x
1
239*x
239*x
388
388
388
150*x
239*x
388
150*x
388
239*x
388
239*x
239*x
150*x
239*x
239*x
1
1
239*x
150*x
1
150*x
150*x
388
1
1
239*x
150*x
388
150*x
239*x
1
1
239*x
1
239*x
150*x
239*x
239*x
388
388
150*x
239*x
1
239*x
239*x
239*x
1
239*x
239*x
150*x
388
239*x
239*x
388
388
1
1
150*x
1
150*x
388
239*x
150*x
388
239*x
239*x
1
388
239*x
239*x
150*x
388
150*x
388
1
239*x
388
150*x
150*x
150*x
1
388
1
388
388
1
1
150*x
150*x
388
239*x
239*x
388
1
239*x
388
388
1
388
1
150*x
239*x
150*x
1
1
1
1
388
150*x
239*x
314*x + 194
239*x
150*x
150*x
1
239*x
239*x
239*x
150*x
1
150*x
1
150*x
1
388
150*x
1
239*x
388
239*x
1
150*x
150*x
150*x
1
1
150*x
150*x
239*x
388
1
150*x
388
388
150*x
1
239*x
150*x
388
388
150*x
239*x
239*x
150*x
1
1
239*x
150*x
150*x
239*x
388
388
239*x
150*x
1
239*x
388
388
239*x
1
388
150*x
239*x
239*x
1
1
239*x
239*x
239*x
1
150*x
388
150*x
1
239*x
388
1
239*x
1
239*x
1
239*x
150*x
150*x
150*x
1
239*x
239*x
150*x
75*x + 194
150*x
239*x
388
1
1
1
1
239*x
150*x
239*x
1
388
1
388
388
150*x
1
388
150*x
150*x
388
239*x
239*x
1
1
388
388
1
388
1
239*x
239*x
239*x
388
150*x
1
388
239*x
388
239*x
150*x
150*x
388
1
150*x
150*x
388
239*x
150*x
388
239*x
1
239*x
1
1
388
388
150*x
150*x
388
239*x
150*x
150*x
1
150*x
150*x
150*x
1
150*x
239*x
388
388
150*x
150*x
239*x
150*x
1
150*x
1
1
150*x
239*x
388
239*x
150*x
1
1
388
239*x
239*x
1
239*x
150*x
1
1
150*x
150*x
239*x
150*x
150*x
388
150*x
388
239*x
388
150*x
239*x
388
388
388
150*x
150*x
1
150*x
388
239*x
1
388
1
1
1
1
388
239*x
1
388
1
388
1
388
239*x
239*x
150*x
239*x
388
239*x
150*x
388
388
150*x
1
150*x
1
150*x
388
388
239*x
388
388
> s := 0; for z in [1..388] do if Coefficient((1+z*x)^((389-1) div 2),1) ne 0 then s:= s +1; end for;

>> 89-1) div 2),1) ne 0 then s:= s +1; end for;
                                           ^
User error: bad syntax
> s := 0; for z in [1..388] do if Coefficient((1+z*x)^((389-1) div 2),1) ne 0 then s:= s +1; end if; end for;
> s;
196
> s*2;
392
> s := 0; for z in [1..388] do if Coefficient((1+z*x)^((389-1) div 2),1) eq 0 then s:= s +1; end if; end for;
> s;
192
> 192+196;
388
> s := 0; for z in [1..388] do if Coefficient((1+z*x)^((389-1) div 2),0) eq 0 then s:= s +1; end if; end for;
> s;
194
> 194 + 192;
386
>