Given a matrix (resp. list of vectors), extend it to a square matrix (resp. list of vectors), such that its determinant is the gcd of its minors (i.e. extend the basis of a lattice to a “maximal” one in Z^n).

Author(s): Gonzalo Tornaria and Jonathan Hanke.

INPUT:

a matrix, or a list of length n vectors (in the same space)

OUTPUT:

a square matrix, or a list of n vectors (resp.)

EXAMPLES:

sage: A = Matrix(ZZ, 3, 2, range(6))
sage: extend_to_primitive(A)
[ 0  1 -1]
[ 2  3  0]
[ 4  5  0]

sage: extend_to_primitive([vector([1,2,3])])
[(1, 2, 3), (0, 1, 1), (-1, 0, 0)]


Return whether n is a triangular number.

A triangular number is a number of the form $$k(k+1)/2$$ for some non-negative integer $$n$$. See Wikipedia article Triangular_number. The sequence of triangular number is references as A000217 in the Online encyclopedia of integer sequences (OEIS).

If you want to get the value of $$k$$ for which $$n=k(k+1)/2$$ set the argument return_value to True (see the examples below).

INPUT:

• n - an integer

• return_value - a boolean set to False by default. If set to True the function returns a pair made of a boolean and the value v such that $$v(v+1)/2 = n$$.

EXAMPLES:

sage: is_triangular_number(3)
True
sage: is_triangular_number(3, return_value=True)
(True, 2)
sage: 2*(2+1)/2
3

sage: is_triangular_number(2)
False
sage: is_triangular_number(2, return_value=True)
(False, None)

sage: is_triangular_number(25*(25+1)/2)
True

sage: is_triangular_number(10^6 * (10^6 +1)/2, return_value=True)
(True, 1000000)

sage: least_quadratic_nonresidue(5)