A341420 The positive integer numbers k represented properly by the binary quadratic form x^2 + 4*y^2.

1, 4, 5, 8, 13, 17, 20, 25, 29, 37, 40, 41, 52, 53, 61, 65, 68, 73, 85, 89, 97, 100, 101, 104, 109, 113, 116, 125, 136, 137, 145, 148, 149, 157, 164, 169, 173, 181, 185, 193, 197, 200, 205, 212, 221, 229, 232, 233, 241, 244, 257, 260, 265, 269, 277, 281, 289, 292, 293, 296

Offset: 1

Author

Wolfdieter Lang, Mar 19 2021

Keywords

• nonn

Comments

If also improper solutions of the Diophantine equation X^2 + Y^2 = k, with positive integer number k are taken into account one can obtain the present solutions provided X or Y are even. E.g., k = 4 has only improper solutions like (X, Y) = (0, pm2) or (pm2, 0) (pm stands for +1 or -1). So 4 is not a member of A008784, but in the present sequence it appears from (x, y) = (0, pm1) obtained from the first (X, Y) solution by y = Y/2.

The number k = 2 = A008784(2) is not represented here because there is only the proper solution (X, Y) = (pm1, pm1).

The number of solutions m(k = a(n)), up to an overall sign change in x and y, is given by m(1) = 1, m(4) = 1, m(8) = 2 and for k = 4^a*8^b*Product_{j=1..P1} (p1_j)^e1_j, with (a,b) from {(0, 0), (1, 0), (0, 1)}}, primes p1_j congruent to 1 (mod 4) (from A002144) and nonnegative exponents e1_j, it is m(k) = 2^(b + P1).

The primitive parallel binary quadratic forms of discriminant -16 = -4*4 representing positive integer numbers k are obtained by solving the Diophantine equation j^2 + 4 == 0 (mod k), for j from {0, 1, ..., k-1}. This gives for k = 1, 2, 4, and 8 the solutions j = 0, 0, {0, 2}, and {2, 6}, respectively. No larger powers of 2 have solutions. No lifting is possible (see Apostol, Theorem 5.30). For odd primes k the Legendre symbol (-4, k) = +1 exactly for k = prime == 1 (mod 4) (from the Legendre symbol (-1, prime) = +1 only for these primes A008784).

These parallel forms are given by (k, 2*j(k), c(j(k))), with c(j(k)) = (j(k)^2 + 4)/k.

There is only one primitive reduced form for discriminant -16, namely the principal form (1, 0, 4) (see the Buell reference p. 20). Thus each parallel form is equivalent (with a determinant +1 transformation) to this principal form, and gives a proper solution.

Examples

			Proper solutions (x, y) (up to overall sign flip) for various k = a(n):
a(2) = 4: (1, 0), m(4) = 1 (a = 1, b = 0, P1 = 0), (2, 0) is not a proper solution);
a(4) = 8: (2, pm1): (pm stands for +1 or -1), m(8) = 2 (a = 0, b = 1, P1 = 0);
a(7) = 20 = 4*5: (4, pm1), m(20) = 2 (a = 1, b = 0, P1 = 1), (m(4) = 1);
a(8) = 25 = 5^2: (3, pm2), m(25) = 2 (a = 0, b = 0, P1 = 1);
a(42) = 200 = 8*5^2: (2, pm7), (14, pm1), m(200) = 4 (a = 0, b = 1, P1 = 1).
		

References

• Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 120-122.
• D. A. Buell, Binary Quadratic Forms, Springer, 1989, p. 20.

Crossrefs

Cf. Discriminants -4: A008784, -8 A057127, -12 A244819.

Formula

a(n) = x(n)^2 + (2*y(n))^2, with gcd(x(n), y(n)) = 1, for n >= 1.