cp's OEIS Frontend

The row length of this irregular triangle is 2*A307372(n).

The indefinite binary quadratic Pell form is F = [1, 0, -D(n)], with D(n) = A000037(n) (D not a square). This form is not reduced (see the Buell or Scholz-Schoeneberg references, and the W. Lang link in A225953 for the definition).

The first reduced form, obtained after two equivalence transformations, is FR(n) = [1, 2*s(n), -(D(n) - s(n)^2)] where s(n) = A000194(n) = D(n) - n, for n >= 1. Hence FR(n) = [1, 2*s(n), -(n - s(n)*(s(n)-1))]. For the two transformations invoving R(t) = matrix([0, -1], [1, t]), first with t = 0 then with t = s(n) see a comment in A000194, and the proposition in the W. Lang link given below. FR(n) is the principal form F_p(n) of discriminant 4*D(n).

Each reduced form FR(n) leads to a cycle of reduced forms with (primitive) period P(n) = 2*p(n) = 2*A307372(n). The sequence of R-transformations is given by the parameter tuple (t_1(n), ..., t_{2*p(n)}(n)) with alternating signs which give the row entries T(n, k) = t_k(n). See also Table 2 of the W. Lang link.

The automorphic transformation is obtained by the matrix Auto(n) = R(t_1(n))*R(t_2(n))*...*R(t_{2*p(n)}(n)). Together with the matrix B(n) := R(0)*R(s(n)) = Matrix([-1, s(n)], [0, -1]) one finds all solutions of the Pell equation x^2 - D(n)*y^2 = +1. For each n >= 1 there is one family (also called class) of proper solutions. The general solution is (x(n;j), y(n;j))^T = B(n)*(Auto(n))^j*(1,0)^T, for integer j (T for transposed). One can always choose x >= 1 by an overall sign flip in x and y.

For the general Pell equation x^2 - D(n)*y^2 = N, for integer N, the parallel forms equivalent to FR(n) become important. For details see the W. Lang link given below, section 3.

For details see A324252 which gives the array for the numbers of families of proper solutions of x^2 - D(n)*y^2 = k for positive integers k. See also the W. Lang link in A324251, section 3.

The D(n) values for nonzero entries in column k = 1 are given in A003814 (representation of -1).

The position list for nonzero entries in row n = 1 is A057126 (conjecture).

The corresponding y values are given in A339882.

The Diophantine equation x^2 - p*y^2 = -2, of discriminant Disc = 4*p > 0 (indefinite binary quadratic form), with prime p can have proper solutions (gcd(x, y) = 1) only for primes p = 2 and p == 3 (mod 8) by parity arguments.

There are no improper solutions (with g >= 2, g^2 does not divide 2).

The prime p = 2 has just one infinite family of proper solutions with nonnegative x values. The fundamental proper solutions for p = 2 is (0, 1).

If a prime p congruent to 3 modulo 8, (p(n) = A007520(n)) has a solution then it can have only one infinite family of (proper) solutions with positive x value.

This family is self-conjugate (also called ambiguous, having with each solution (x, y) also (x, -y) as solution). This follows from the fact that there is only one representative parallel primitive form (rpapf), namely F_{pa(n)} = [-2, 2, -(p(n) - 1)/2].

The reduced principal form of Disc(n) = 4*p(n) is F_{p(n)} = [1, 2*s(n), -(p(n) - s(n)^2)], with s(n) = A000194(n'(n)), if p(n) = A000037(n'(n)). The corresponding (reduced) principal cycle has length L(n) = 2*A307372(n'(n)).

The number of all cycles, the class number, for Disc(n) is h(n'(n)) = A324252(n'(n)), Note that in the Buell reference, Table 2B in Appendix 2, p. 241, all Disc(n) <= 4*1051 = 4204 have class number 2, except for p = 443, 499, 659 (Disc = 1772, 1996, 26).

See the W. Lang link Table 1 for some principal reduced forms F_{p(n)} (there for p(n) in the D-column, and F_p is called FR(n)) with their t-tuples, giving the automporphic matrix Auto(n) = R(t_1) R(t_1) ... R(t_{L(n)}), where R(t) := Matrix([[0, -1], [1, t]]), and the length of the principal cycle L(n) given above, and in Table 2 for CR(n).

To prove the existence of a solution one would have to show that the rpapf F{pa(n)} is properly equivalent to the principal form F_{pa(n)}.

For the definition of representative parallel primitive forms (rpapfs) for discriminant Disc > 0 (the indefinite case) and representation of nonzero integers k see the Scholz-Schoeneberg reference, p. 105, or the Buell reference p. 49 (without use of the name parallel). For the procedure to find the primitive representative parallel forms (rpapfs) for Disc(n) = 4*D(n) = 4*A000037(n) and nonzero integer k see the W. Lang link given in A324251, section 3.

Note that the number of rpapfs of a discriminant Disc > 0 for k >= 1 is identical with the one for negative k. These forms differ in the signs of the a and c entries of these forms but not the b >= 0 entry (called an outer sign flip). See some examples below, and the program in the mentioned W. Lang link, section 3.

For the forms counted in the array A(n, k) see Table 3 of the W. Lang link given in A324251, for n = 1..30 and k = 1..10.

Compare the present array with the ones given in A324252 and A307303 for the number of rpapfs for discriminant 4*D(n) and representable positive and negative k, respectively, that are equivalent (under SL(2, Z)) to the reduced principal form F_p = [1, 2*s(n), -(D(n) - s(n)^2)] with s(n) = A000194(n), of the unreduced Pell form F(n) = [1, 0, -D(n)].

The rpapfs not counted in A324252 and A307303 are equivalent to forms of non-principal cycles for discriminant 4*D(n).

The total number of cycles (the class number h(n)) for discriminant 4*D(n) is given in A307359(n).

The array for the length of the periods of these cycles is given in A307378.

One half of the sum of the length of the periods is given in A307236.