cp's OEIS Frontend

A084917 Positive numbers of the form 3*y^2 - x^2.

%I A084917 #75 Feb 20 2024 02:33:54
%S A084917 2,3,8,11,12,18,23,26,27,32,39,44,47,48,50,59,66,71,72,74,75,83,92,98,
%T A084917 99,104,107,108,111,122,128,131,138,143,146,147,156,162,167,176,179,
%U A084917 183,188,191,192,194,200,207,218,219,227,234,236,239,242,243,251,263,264,275,282,284
%N A084917 Positive numbers of the form 3*y^2 - x^2.
%C A084917 Positive integers k such that x^2 - 4xy + y^2 + k = 0 has integer solutions. (See the CROSSREFS section for sequences relating to solutions for particular k.)
%C A084917 Comments on method used, from _Colin Barker_, Jun 06 2014: (Start)
%C A084917 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.
%C A084917 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.
%C A084917 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:
%C A084917 "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."
%C A084917 Incidentally all my Pell code is written in B-Prolog, and is somewhat voluminous. (End)
%C A084917 Also, positive integers of the form -x^2 + 2xy + 2y^2 of discriminant 12. - _N. J. A. Sloane_, May 31 2014 [Corrected by _Klaus Purath_, May 07 2023]
%C A084917 The equivalent sequence for x^2 - 3xy + y^2 + k = 0 is A031363.
%C A084917 The equivalent sequence for x^2 - 5xy + y^2 + k = 0 is A237351.
%C A084917 A positive k does not appear in this sequence if and only if there is no integer solution of x^2 - 3*y^2 = -k with (i) 0 < y^2 <= k/2 and (ii) 0 <= x^2 <= k/2. See the Nagell reference Theorems 108 a and 109, pp. 206-7, with D = 3, N = k and (x_1,y_1) = (2,1). - _Wolfdieter Lang_, Jan 09 2015
%C A084917 From _Klaus Purath_, May 07 2023: (Start)
%C A084917 There are no squares in this sequence. Products of an odd number of terms as well as products of an odd number of terms and any terms of A014209 belong to the sequence.
%C A084917 Products of an even number of terms are terms of A014209. The union of this sequence and A014209 is closed under multiplication.
%C A084917 A positive number belongs to this sequence if and only if it contains an odd number of prime factors congruent to {2, 3, 11} modulo 12. If it contains prime factors congruent to {5, 7} modulo 12, these occur only with even exponents. (End)
%C A084917 From _Klaus Purath_, Jul 09 2023: (Start)
%C A084917 Any term of the sequence raised to an odd power also belongs to the sequence. Proof: t^(2n+1) = t*t^2n = (3*x^2 - y^2)*t^2n = 3*(x*t^n)^2 - (y*t^n)^2. It seems that t^(2n+1) occurs only if t also is in the sequence.
%C A084917 Joerg Arndt has proved that there are no squares in this sequence: Assume s^2 = 3*y^2 - x^2, then s^2 + x^2 = 3 * y^2, but the sum of two squares cannot be 3 * y^2, qed. (End)
%C A084917 That products of any 3 terms belong to the sequence can be proved by the following identity:  (na^2 - b^2) (nc^2 - d^2) (ne^2 - f^2) = n[a(nce + df) + b(cf + de)]^2 - [na(cf + de) + b(nce + df)]^2. This can be verified by expanding both sides of the equation. - _Klaus Purath_, Jul 14 2023
%D A084917 T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.
%H A084917 Sci.math, <a href="http://www.math.niu.edu/~rusin/known-math/98/pells">General Pell equation: x^2 - N*y^2 = D</a>, 1998
%H A084917 Sci.math, <a href="http://oeis.org/A035251/a035251.txt">General Pell equation: x^2 - N*y^2 = D</a>, 1998 (Edited and cached copy)
%H A084917 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%e A084917 11 is in the sequence because 3 * 3^2 - 4^2 = 27 - 16 = 11.
%e A084917 12 is in the sequence because 3 * 4^2 - 6^2 = 48 - 36 = 12.
%e A084917 13 is not in the sequence because there is no solution in integers to 3y^2 - x^2 = 13.
%e A084917 From _Wolfdieter Lang_, Jan 09 2015: (Start)
%e A084917 Referring to the Jan 09 2015 comment above.
%e A084917 k = 1 is out because there is no integer solution of (i) 0 < y^2 <= 1/2.
%e A084917 For k = 4, 5, 6, and 7 one has y = 1, x = 0, 1 (and the negative of this). But x^2 - 3 is not -k for these k and x values. Therefore, these k values are missing.
%e A084917 For k = 8 .. 16 one has y = 1, 2 and x = 0, 1, 2. Only y = 2 has a chance and only for k = 8, 11 and 12 the x value 2, 1 and 0, respectively, solves x^2 - 12 = -k. Therefore 9, 10, 13, 14, 15, 16 are missing.
%e A084917 ... (End)
%t A084917 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 *)
%t A084917 Select[Range[300],Length[FindInstance[3y^2-x^2==#,{x,y},Integers]]>0&] (* _Harvey P. Dale_, Apr 23 2023 *)
%Y A084917 With respect to solutions of the equation in the early comment, see comments etc. in: A001835 (k = 2), A001075 (k = 3), A237250 (k = 11), A003500 (k = 12), A082841 (k = 18), A077238 (k = 39).
%Y A084917 Cf. A031363, A237351.
%Y A084917 A141123 gives the primes.
%Y A084917 For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
%K A084917 nonn,easy
%O A084917 1,1
%A A084917 _Roger Cuculière_, Jul 14 2003
%E A084917 Terms 26 and beyond from _Colin Barker_, Feb 06 2014