A378710 Positive numbers k such that -k is properly represented by the Pell Form x^2 - 15*y^2.

6, 11, 14, 15, 35, 51, 59, 71, 86, 110, 119, 131, 134, 159, 179, 191, 206, 215, 231, 239, 251, 254, 294, 311, 326, 335, 339, 359, 366, 371, 374, 411, 419, 431, 446, 479, 491, 519, 515, 519, 539, 566, 590, 591, 599, 614, 635, 654, 659, 671, 686

Offset: 1

Author

Wolfdieter Lang, Dec 13 2024

Keywords

• nonn

Comments

This is a subsequence of A237606. There the uninteresting numbers that have improper representations are also recorded.

The primes in the sequence are given in A141302.

A primitive indefinite form F(a,b,c;x,y) = a*x^2 + b*x*y + c*y^2, or [a, b, c] with gcd(a, b, c) = 1 and even discriminant Disc = b^2 - 4*a*c = 60 = 4*D, D = 15, has class number A307359(12) = 4. The four reduced 2-cycle forms are the principal cycle CR = {[1, 6, -6], [6, 6,-1]}, CRhat = {[-1, 6, 6], [-6, 6,1]}, (outer signs flipped), Cy ={[2, 6, -3], [-3, 6, 2]} and Cyhat ={[-2, 6, 3], [3, 6, -2]}.

A proper representation of an integer k (not 0) by such a form F is determined by the rpapfs (representative parallel primitive forms) FPa(k, j) = [k, 2*j, (j^2 - 15)/k], where j from {0, 1, ...,|k|-1} is determined by the congruence j^2 - 15 = = 0 (mod |k|).

The equivalence transformations R(t) of a form F = [a, b, c] is [c , -b +2*c*t, 1 - b*t + c*t^2]. This corresponds to R(t) = Matrix([0, -1], [1, t]). Half-reduced R-transformations use the choice t = ceiling((8 + b)/(2*c) - 1), if c > 0, and t = floor(1 - (8 + b)/(2*|c|)) if c < 0. (c = 0 is not considered because Disc becomes a square).

Because any form F of Disc = 60 represents a negative integer -k if it is equivalent to one of the rpapfs FPa(-k, j), the allowed values are

k = 2^{e_2}*3^{e_3}*5^{e_5}*Product_{i=1..P} p_i^{e_j}, where p_i is an odd prime >= 7 from the sequence A097956 or A038887(n), n >= 4, the p with Legendre(15, p) = +1. The exponents for 2, 3, and 5 are from {0, 1} (these primes are not liftable to powers) and e_i >= 0 (p_i is uniquely liftable to powers, see the Apostol reference), but not all exponents should be 0, because -1 is not represented. The number of infinite families of proper solutions (x, y), with positive values y, is 2^(P).

The present sequence is a proper subset of these generally allowed k values. One has to check if the rpapfs Fpa(-k, j) reach the principal cycle CR, then if so k is a member of the present sequence. This is because the Pell form FPell = [1, 0, -15] reaches (taking t to be first 0 then 3) the cycle member CR(1) = [1, 6, -6], the reduced principal form.

For details see the W. Lang paper in the links.

For the fundamental proper positive solutions of the infinite families for - a(n) see A378711. Note that -a(n) may also have improper solutions besides the proper ones whenever even powers of primes satisfying Legendre(15, p) = +1 appear, e.g., the first instance being -294 = -2*3*7^2).

Examples

			-2, -3, and -5 are not in the sequence because the rpapfs are [-2, 2, 7] reaching after two R(t)-steps with t values -0 and  -1 the cycle member Cyhat(1), [-3, 0, 5] reaching with t values 0  and 1 Cy(1), and [-5, 0, 3] reaches with t = 0 Cyhat(2), respectively.
-a(1) = -6 = -2*3 is represented because [-6, 6, 1] = CR(2) (already a reduced form). There is only one infinite family of proper solutions with y > 0 (an ambiguous case) with fundamental solution (x, y) = (3, 1).
There is no solution representing  -10 = -2*5, because [-10, 10, -1] leads with t = -8 to CRhat(1).
-a(11) = - 119 has the four rpapfs [-119, 54, -6], [-119, 82, -14], [-119, 156, -51], and [-119, 184, -71]. They lead with t = -5,  t = -3, 4, t = -1, 2, 2, and t = -1, 3 to members CR(2), CR(1), CR(1), and CR(2), respectively.
		

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986. Theorem 5.10, pp, 121-122.
• A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 21 - 34.
• Trygve Nagell, Introduction to Number Theory, 2nd edition, Chelsea Publishing Company, 1964, pp. 195 - 212.
• A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, chapter IV, pp. 97 - 126.

Links

• Wolfdieter Lang, Cycles of reduced Pell forms, general Pell equations and Pell graphs

Crossrefs

Cf. A038887, A097956, A141302, A237606, A307359, A378711.