A089270 Positive numbers represented by the integer binary quadratic form x^2 + x*y - y^2 with x and y relatively prime.

1, 5, 11, 19, 29, 31, 41, 55, 59, 61, 71, 79, 89, 95, 101, 109, 121, 131, 139, 145, 149, 151, 155, 179, 181, 191, 199, 205, 209, 211, 229, 239, 241, 251, 269, 271, 281, 295, 305, 311, 319, 331, 341, 349, 355, 359, 361, 379, 389, 395, 401, 409, 419, 421, 431

Offset: 1

Author

Wolfdieter Lang, Nov 07 2003

Keywords

• nonn

Comments

The negative numbers represented by x^2 + x*y - y^2 with relative prime x and y are -a(n).

The discriminant of this binary form is D = 5 > 0, hence this is an indefinite form.

It appears that these are also the numbers k for which the equation x^2 = x+1 (mod k) has solutions. The number of solutions is 0 or a power of 2. It appears that k=5 is the only k for which x^2 = x+1 (mod k) has just one solution. The first k producing 4 solutions is 209. The first k producing 8 solutions is 6061. - T. D. Noe, Nov 04 2009 [For a proof see the W. Lang link, Proposition. - Wolfdieter Lang, Jul 04 2019]

(Conjecture) The terms are the products of primes congruent to {0,1,4} mod 5 (using at most a single 5, but repeating other primes is allowed), which is A038872. - T. D. Noe, Nov 14 2010 [Comment in brackets from Shreevatsa R, Mar 27 2019. For a proof see the W. Lang link, Lemma 1, iii) and the Proposition. - Wolfdieter Lang, Jul 04 2019]

Brousseau's paper lists these numbers (less than 1000) as discriminants of Fibonacci sequences. For each number, he also lists the (a,b) pairs that are the first two terms of a unique Fibonacci sequence. [These numbers are not discriminants, which is evident from the fact that not all of them are congruent to 0 or 1 modulo 4. Although Brousseau denotes them with D, he calls them "quantity ... which is characteristic of any given sequence". The same list can be found in the letter to N. J. A. Sloane by Hoggatt, Jr. where the numbers are called "characteristic numbers of Fibonacci sequences". Finally, Matthew Staller in his comment below calls them "determinants", which is probably the most appropriate term. - Klaus Purath, Sep 08 2022]

From Matthew Staller, Oct 01 2015: (Start)

The number of fundamental solutions to n = |y^2 - x^2 - x*y| with relatively prime x and y is 0 or 2^k, where k is the number of distinct prime factors of n that are congruent to {1,4} mod 5 (conjecture). For example, n=187891=11*19*29*31 has 16=2^4 solutions; the prime n=9999999929 has 2=2^1 solutions; n=84182245951=31^3*41^4 has 4=2^2 solutions. [For a proof of the conjecture see the W. Lang link, Lemma 1, iii) and the Proposition. - Wolfdieter Lang, Jul 04 2019]

Recurring sequences (as Fibonacci sequences) may be ordered by determinant (|y^2 - x^2 - x*y| for consecutive (x,y) terms), and further by individual terms to clarify where necessary. For example, the four distinct sequences that have a determinant of 209 are (13,8), (13,5), (14,13), (14,1), which shows how they were found but which would be more commonly understood as (8,21), (5,18), (13,27), (1,15). For a determinant of 1 there is exactly one sequence (Fibonacci, A000045); for a determinant of 5 there is just one (Lucas, A000032). For 11 there are two (1,4) and (2,5), the latter of which is known as the Evangelist Sequence (A001060).

(End)

The linear map (x,y) -> (5x+8y, 8x+13y) maps coprime integer solutions of x^2 + x*y - y^2 = n to coprime integer solutions, so if there is such a solution with nonnegative x,y there must be one with y < 8*sqrt(n). - Robert Israel, Oct 01 2015

Odd numbers k such that 5 is a square mod k. - Shreevatsa R, Mar 27 2019 [For a proof see the W. Lang link, Lemma 1, iii) and Proposition. - Wolfdieter Lang, Jul 04 2019]

Let m = a^2 + a*b - b^2 and n = c^2 + c*d - d^2, where gcd(a, b) = gcd(c, d) = 1. If a*d - b*c = 1, then A165900(a*c + a*d - b*d) = m*n. - Isaac Saffold, Feb 23 2020

Examples

			n=2: a(2)=5 with, for example, (x,y)= (2,1): 4+2-1=5 (there are infinitely many proper (x,y) solutions).
n=8: a(8)=55 with, for example, (x,y)=(7,6) or (7,1). In this case there exist two fundamental proper solutions.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000, matches Jagy's program output (_R. J. Mathar_, Sep 10 2016)
• Alfred Brousseau, On the ordering of Fibonacci sequences, Fib. Quart. 1.4 (1963), 43-46; Errata, Fib. Quart. 2.1 (1964), 38.
• V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
• Norbert Hungerbühler and Maciej Smela, Geometric approach to the Diophantine equation x^2 + x*y - y^2 = m, hal-04835410, 2024. See p. 18.
• Wolfdieter Lang, Fibonacci sequences with relative prime initial conditions, and the binary quadratic form [1, 1, -1]
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Odd numbers in A057762.

Disjoint union of A336403 and 5*A336403.

Programs

Maple

F:= proc(n) local x,y;
      for y from 1 to floor(8*sqrt(n)) do
         x := (-y+sqrt(5*y^2+4*n))/2;
         if x::integer and igcd(x,y) = 1 then return true fi;
      od:
      false
end proc:
select(F, [$1..1000]); # Robert Israel, Oct 01 2015

Mathematica

Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + x y - y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Print[n]; Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

PARI

for (k=1, 431, if(#qfbsolve(Qfb(1,1,-1),factor(k),1), print1(k,", "))) \\ Hugo Pfoertner, Sep 09 2022

Formula

a(n) = x^2 + x*y - y^2 with relatively prime integers x and y (proper solutions of the Diophantine equation).

Extensions

Minor edits by Matthew Staller, Jun 05 2019