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A084917 Positive numbers n such that n = 3*y^2 - x^2 with integers x, y. 10
2, 3, 8, 11, 12, 18, 23, 26, 27, 32, 39, 44, 47, 48, 50, 59, 66, 71, 72, 74, 75, 83, 92, 98, 99, 104, 107, 108, 111, 122, 128, 131, 138, 143, 146, 147, 156, 162, 167, 176, 179, 183, 188, 191, 192, 194, 200, 207, 218, 219, 227, 234, 236, 239, 242, 243, 251, 263, 264, 275, 282, 284 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Positive integers k such that x^2 - 4xy + y^2 + k = 0 has integer solutions.

Comments on method used, from Colin Barker, Jun 06 2014: (Start)

In general, we want to find the values of f, from 1 to 400 say, for which x^2 + bxy + y^2 + f = 0 has integer solutions for a given b.

In order to solve x^2 + bxy + y^2 + f = 0 we can solve the Pellian equation x^2 - Dy^2 = N, where D = (b*b - 4) and N = 4*(b*b - 4)*f.

But since sqrt(D) < N, the classical method of solving x^2 - Dy^2 = N does not work. So I implemented the method described in the 1998 sci.math reference, which says:

"There are several methods for solving the Pellian equation when |N| > sqrt(d). One is to use a brute-force search. If N < 0 then search on y = sqrt(abs(n/d)) to sqrt((abs(n)(x1 + 1))/(2d)) and if N > 0 search on y = 0 to sqrt((n(x1 - 1))/(2d)) where (x1, y1) is the minimum positive solution (x, y) to x^2 - dy^2 = 1.  If N < 0, for each positive (x, y) found by the search, also take (-x, y).  If N > 0, also take (x, -y).  In either case, all positive solutions are generated from these using (x1, y1) in the standard way."

Incidentally all my Pell code is written in B-Prolog, and is somewhat voluminous. (End)

Also, positive integers of the form -x^(+2xy) + 2y^2 of discriminant 12. - N. J. A. Sloane, May 31 2014

The equivalent sequence for x^2 - 3xy + y^2 + k = 0 is A031363.

The equivalent sequence for x^2 - 5xy + y^2 + k = 0 is A237351.

A positive n does not appear in this sequence if and only if there is no integer solution of x^2 - 3*y^2 = -n with  i) 0 < y^2 <= n/2 and ii) 0 <= x^2 <= n/2. See the Nagell reference Theorems 108 a  and 109, pp. 206-7, with D = 3, N = n and (x_1,y_1) = (2,1). - Wolfdieter Lang, Jan 09 2015

REFERENCES

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

LINKS

Table of n, a(n) for n=1..62.

Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998

Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998 (Edited and cached copy)

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

EXAMPLE

11 is in the sequence because 3 * 3^2 - 4^2 = 27 - 16 = 11.

12 is in the sequence because 3 * 4^2 - 6^2 = 48 - 36 = 12.

13 is not in the sequence because there is no solution in integers to 3y^2 - x^2 = 13.

From Wolfdieter Lang, Jan 09 2015: (Start)

Referring to the Jan 09 2015 comment above.

n = 1 is out because there is no integer solution of i) 0 < y^2 <= 1/2.

For n = 4, 5, 6, and 7 one has y = 1, x = 0, 1 (and the negative of this). But x^2 - 3 is not -n for these n and x values. Therefore, these n values are missing.

For n = 8 .. 16 one has y = 1, 2 and x = 0, 1, 2. Only y = 2 has a chance and only for n = 8, 11 and 12 the x value 2, 1 and 0, respectively, solves x^2 - 12 = -n. Therefore 9, 10, 13, 14, 15, 16 are missing.

... (End)

MATHEMATICA

r[n_] := Reduce[n == 3*y^2 - x^2 && x > 0 && y > 0, {x, y}, Integers]; Reap[For[n = 1, n <= 1000, n++, rn = r[n]; If[rn =!= False, Print["n = ", n, ", ", rn /. C[1] -> 1 // Simplify]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 21 2016 *)

CROSSREFS

Cf. A001835 (k = 2), A001075 (k = 3), A237250 (k = 11), A003500 (k = 12), A082841 (k = 18), A077238 (k = 39).

Cf. A031363, A237351.

A141123 gives the primes.

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Sequence in context: A005727 A118089 A201541 * A134713 A293910 A173269

Adjacent sequences:  A084914 A084915 A084916 * A084918 A084919 A084920

KEYWORD

nonn,easy

AUTHOR

Roger Cuculière, Jul 14 2003

EXTENSIONS

Terms 26 and beyond from Colin Barker, Feb 06 2014

STATUS

approved

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Last modified February 15 16:53 EST 2019. Contains 320136 sequences. (Running on oeis4.)