Function: issquare
Section: number_theoretical
C-Name: issquareall
Prototype: lGD&
Help: issquare(x,{&n}): true(1) if x is a square, false(0) if not. If n is
given puts the exact square root there if it was computed.
Description:
(int):bool Z_issquare($1)
(gen):bool issquare($1)
(int, &int):bool Z_issquareall($1, &$2)
(gen, &gen):bool issquareall($1, &$2)
Doc: true (1) if $x$ is a square, false (0)
if not. What ``being a square'' means depends on the type of $x$: all
\typ{COMPLEX} are squares, as well as all non-negative \typ{REAL}; for
exact types such as \typ{INT}, \typ{FRAC} and \typ{INTMOD}, squares are
numbers of the form $s^2$ with $s$ in $\Z$, $\Q$ and $\Z/N\Z$ respectively.
\bprog
? issquare(3) \\ as an integer
%1 = 0
? issquare(3.) \\ as a real number
%2 = 1
? issquare(Mod(7, 8)) \\ in Z/8Z
%3 = 0
? issquare( 5 + O(13^4) ) \\ in Q_13
%4 = 0
@eprog
If $n$ is given, a square root of $x$ is put into $n$.
\bprog
? issquare(4, &n)
%1 = 1
? n
%2 = 2
@eprog
For polynomials, either we detect that the characteristic is 2 (and check
directly odd and even-power monomials) or we assume that $2$ is invertible
and check whether squaring the truncated power series for the square root
yields the original input.
For \typ{POLMOD} $x$, we only support \typ{POLMOD}s of \typ{INTMOD}s
encoding finite fields, assuming without checking that the intmod modulus
$p$ is prime and that the polmod modulus is irreducible modulo $p$.
\bprog
? issquare(Mod(Mod(2,3), x^2+1), &n)
%1 = 1
? n
%2 = Mod(Mod(2, 3)*x, Mod(1, 3)*x^2 + Mod(1, 3))
@eprog
Variant: Also available is \fun{long}{issquare}{GEN x}. Deprecated
GP-specific functions \fun{GEN}{gissquare}{GEN x} and
\fun{GEN}{gissquareall}{GEN x, GEN *pt} return \kbd{gen\_0} and \kbd{gen\_1}
instead of a boolean value.