A378711 Irregular triangle read by rows: row n gives the proper positive integer fundamental solutions (x, y) of x^2 - 15*y^2 = - A378710(n), for n >= 1.

3, 1, 2, 1, 7, 2, 1, 1, 11, 3, 15, 4, 5, 2, 10, 3, 3, 2, 18, 5, 1, 2, 26, 7, 8, 3, 13, 4, 7, 3, 17, 5, 5, 3, 25, 7, 4, 3, 11, 4, 16, 5, 29, 8, 2, 3, 37, 10, 1, 3, 41, 11, 9, 4, 24, 7, 14, 5, 19, 6, 7, 4, 32, 9, 13, 5, 23, 7, 5, 4, 40, 11, 3, 4, 12, 5, 27, 8, 48, 13, 1, 4, 56, 15

Offset: 1

Author

Wolfdieter Lang, Dec 13 2024

Keywords

• nonn
• tabf

Comments

The number of (x, y) pairs in the rows are: 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, ...

For details on the general proper representations of a negative integer k by forms with discriminant Disc = 60 = 4*15 see A378710, with references. For the Pell case x^2 - 15*y^2 only a subset of these k values is permitted, namely those that have representative reduced primitive forms (rpapfs) Fpa(-k, j) (see A378710) equivalent to the principal reduced form CR(1) = [1, 6, -6], which is in turn equivalent to the Pell form FPell = [1, 0, -15].

Some rules for the represented -A378710(n) values are: the negative of the prime factors 2, 3 and 5 of 15 are not represented, they are equivalent to some of the other three 2-cycles forms. Powers of these three primes can never occur because they cannot be lifted (see the Apostol reference, Theorem 5.20, pp. 121-122). The products -2*3 and -3*5 are represented but not -2*5 (the rpapf [-10, 10, -1] is equivalent to [-1, 6, 6], a member of the 2-cycle called CRhat in A378710). -2*3*5 is also not represented ([-30, 30, -7] is equivalent to [2, 6, -3] from the cycle Chat).

The reservoir for the odd primes >= 7 is given by Legendre(15, p) = +1 (A097956 or A038887(n), n >= 4). These primes can be lifted uniquely, but one has to find out for each case, also for the product of powers of these primes (together with 2, 3, and 5 factors) if the rpapfs reach the fundamental cycle CR.

The number of infinite families of proper solutions for k = -A378710(n) with positive y is determined by 2^P(n), where P(n) is the number of primes >= 7 in A378710(n). These numbers 2^P(n) are given in the table below, and in the first comment.

The proper family of solutions {(x(n,i), y(n,i))}_{i = -infinity ... +infinity} are found from the rpapf(-A378710(n), j) = [-A378710(n), 2*j, (15 - j^2)/A378710(n)] with the help of the formula (x(n, i), y(n, i))^T = (+ or - B15)*(-Auto15)^i*Rtvalues(n,j)^(-1)*(1, 0)^T, for the solutions of j^2 - 15 == 0 (mod(A378710(n)), for j from 0, 1,.., A378710(n) - 1, (T for transpose) where B15 = R(0)*R(3) = -Matrix([1, 3], [0, 1]), Auto15 = R(-1)*R(6) = - Matrix([1, 6], [1, 7]). For the R(t)-transformation matrix see A378710(n). Rtvalues(n,j) is the product of R(t) matrices with the t-values leading from the rpapf(-A378710(n), j) to the form CR(1). The sign of B15 is chosen such that no negative values for y appear.

The powers (- Auto15)^i = Matrix([S(i, 8) - 7*S(i-1, 8), 6*S(i-1, 8)], [S(-i, 8), S(i, 8) - S(i-1, 8)]), with the Chebyshev polynomial S(i, 8) given, for i >= -1, in A001090(i) and S_(-i, 8) = -S(i-2, 8), for i >= 2.

Examples

			n,  A378710(n) \  k   1  2    3   4    5  6    7  8       pairs = 2^P
----------------------------------------------------------------------
1,    6 = 2*3      |  3  1                                    1
2,   11            |  2  1,   7   2                           2
3,   14 = 2*7      |  1  1,  11   3                           2
4,   15 = 3*5      | 15  4                                    1
5,   35 = 5*7      |  5  2,  10   3                           2
6,   51 = 3*17     |  3  2,  18   5                           2
7,   59            |  1  2,  26   7                           2
8,   71            |  8  3,  13   4                           2
9,   86 = 2*43     |  7  3,  17   5                           2
10, 110 = 2*5*11   |  5  3,  25   7                           2
11  119 = 7*17     |  4  3,  11   4,  16  5,  29  8           4
12, 131            |  2  3,  37  10                           2
13, 134 = 2*67     |  1  3,  41  11                           2
14, 159 = 3*53     |  9  4,  24   7                           2
15, 179            | 14  5,  19   6                           2
16, 191            |  7  4,  32   9                           2
17, 206 = 2*103    | 13  5,  23   7                           2
18, 215 = 5*43     |  5  4,  40  11                           2
19, 231 = 3*7*11   |  3  4,  12   5,  27  8,  48 13           4
20, 239            |  1  4,  56  15                           2
...
For the representation of -A378710(19) = -231 = -3*7*11 see the linked Figure of the directed and weighted Pell cycle graph with the two pairs of conjugate rpapfs (corresponding to solution of the congruence j^2 - 15 = = 0 (mod 231) with j and 231 - j, for j = 57 and j = 90. There the t-values are given as weights. E.g., the rpapf Fpa4 = [-231. 282, -86] has t-values (1-, 2, 2, 6). The pairs of row n = 19 belong to FPa1, FPa3, Fpa4 and FPa2, with the i exponents in the formula above  0, 0, 1, 1, respectively, and the sign of B15 is - in all four cases.
		

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.

Links

• Wolfdieter Lang, Pell cycle graph for discriminant 60 representing -231.

Crossrefs

Cf. A001090, A097956, A038887, A141302, A378710.