A232550 Number of distinct primitive quadratic forms of discriminant = -4n that exist such that every prime p for which p is a quadratic residue (mod 4n) or p-n is a quadratic residue (mod 4n) can be represented by one of them.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 5, 1, 2, 3, 2, 2, 2, 2, 2, 3, 2, 1, 4, 1, 2, 3, 2, 2, 2, 1, 3, 2, 2, 2, 5, 2, 1, 3, 2, 1, 4, 2, 2, 2, 1, 3, 3, 2, 2, 3, 2, 2

Offset: 1

Author

V. Raman, Nov 26 2013

Keywords

• nonn
• uned

Comments

A quadratic form is primitive if the GCD of the coefficients is 1. For example, the quadratic form 2*x^2+4*y^2 is not primitive.

Two quadratic forms f(x,y) = a*x^2+b*x*y+c*y^2 and g(x,y) = p*x^2+q*x*y+r*y^2 are distinct (or inequivalent) if and only if one cannot be obtained by a linear transformation (of the variables x, y) from the other. For example, the three quadratic forms u(x,y) = 3*x^2+2*x*y+3*y^2, v(x,y) = 3*x^2+4*x*y+4*y^2 and w(x,y) = 3*x^2+10*x*y+11*y^2 are equivalent because v(x,y) = u(x+y,-y) and w(x,y) = v(x+y,y). Also, w(x,y) = u(x+2*y,-y). Similarly, the two quadratic forms s(x,y) = 8*x^2+9*y^2 and t(x,y) = 17*x^2+50*x*y+41*y^2 are equivalent because t(x,y) = s(x+2*y,x+y).

The quadratic form x^2+n*y^2 is one such form and the only such form if n is a convenient number (A000926).

a(n) = 1 if and only if n is a convenient number (A000926).

Any prime p such that p is a quadratic residue (mod 4n) or p-n is a quadratic residue (mod 4n) can be represented by exactly one of the a(n) distinct primitive quadratic forms of discriminant = -4n in at most four different ways (if n >= 2) or in at most eight different ways (if n = 1).

If a prime p can be written in the form x^2+n*y^2, then either p is a quadratic residue (mod 4n) or p-n is a quadratic residue (mod 4n), assuming that p^2 does not divide n.

For primes p such that p is a quadratic residue (mod 4n) or p-n is a quadratic residue (mod 4n), there is a lowest square m^2 such that m^2*p can be written in form x^2+n*y^2, where x and y are nonnegative integers (see A232529 and A232530).

If n is a prime congruent to 3 (mod 4), then a(n) = A232551(n).

The product of two numbers (prime or composite, same or different) which can be represented by the same quadratic form of discriminant = -4n can be written in the form x^2+n*y^2, as the following identity shows.

(X*a^2+Y*a*b+Z*b^2)*(X*c^2+Y*c*d+Z*d^2) = (a*c*X+b*d*Z+a*d*(Y/2)+b*c*(Y/2))^2 + ((X*Z)-(Y^2/4))*(a*d-b*c)^2.

(X*a^2+Y*a*b+Z*b^2)*(X*c^2+Y*c*d+Z*d^2) = (a*c*X+b*d*((Y^2/(2*X))-Z)+a*d*(Y/2)+b*c*(Y/2))^2 + ((X*Z)-(Y^2/4))*(b*d*(Y/X)+a*d+b*c)^2.

Note that for the latter equation, (a*c*X+b*d*((Y^2/(2*X))-Z)+a*d*(Y/2)+b*c*(Y/2)) and (b*d*(Y/X)+a*d+b*c) need not always be integers. If they are both integers, then it will be a second representation of the product of (X*a^2+Y*a*b+Z*b^2) and (X*c^2+Y*c*d+Z*d^2) in the form x^2+((X*Z)-(Y^2/4))*y^2.

Examples

			If n is a convenient number (A000926), then the only such available quadratic form is x^2+n*y^2.
For n = 11, every prime that is congruent to {0, 1, 3, 4, 5, 9} mod 11 can be represented by either of the two distinct primitive quadratic forms of discriminant = -44: x^2+11*y^2 or 3*x^2+2*x*y+4*y^2.
For n = 14, every prime that is congruent to {1, 2, 7, 9, 15, 23, 25, 39} mod 56 can be represented by either of the two distinct primitive quadratic forms of discriminant = -56: x^2+14*y^2 or 2*x^2+7*y^2.
For n = 17, every prime that is congruent to {1, 2, 9, 13, 17, 21, 25, 33, 49, 53} mod 68 can be represented by either of the two distinct primitive quadratic forms of discriminant = -68: x^2+17*y^2 or 2*x^2+2*x*y+9*y^2.
For n = 19, every prime that is congruent to {0, 1, 4, 5, 6, 7, 9, 11, 16, 17} mod 19 can be represented by either of the two distinct primitive quadratic forms of discriminant = -76: x^2+19*y^2 or 4*x^2+2*x*y+5*y^2.
For n = 20, every prime that is congruent to {1, 5, 9} mod 20 can be represented by either of the two distinct primitive quadratic forms of discriminant = -80: x^2+20*y^2 or 4*x^2+5*y^2.
		

Links

• V. Raman, Illustration of above mentioned quadratic forms for n = 1..1000

Crossrefs

Cf. A000003, A000926, A232529, A232530, A232551 (Number of distinct primitive quadratic forms of discriminant = -4*n needed to generate all primes p for which -n is a quadratic residue (mod p)).