A281387 Pairs (x, y) of relatively prime positive integers such that (x^2 - 5)/y and (y^2 - 5)/x are both positive integers.

4, 11, 11, 29, 29, 76, 76, 199, 199, 521, 521, 1364, 1364, 3571, 3571, 9349, 9349, 24476, 24476, 64079, 64079, 167761, 167761, 439204, 439204, 1149851, 1149851, 3010349, 3010349, 7881196, 7881196, 20633239, 20633239, 54018521, 54018521, 141422324

Offset

1,1

Comments

For x, y > 2, the solutions start with (4,11) -> (11, 29) -> (29, 76) -> ...

The sequence is infinite (see the proof in the second reference).

Consider the pairs of the form (a(2n-1), a(2n)). Limit_{n->oo} a(2n)/a(2n-1) = phi^2 = 2.618033988749894... (A104457).

Property: a(2n-1)^2 + a(2n)^2 = 3*a(2n-1)*a(2n) + 5.

Apparently a(2*n) = a(2*n+1) = A002878(n) for n >= 1. - Georg Fischer, Dec 05 2022

Links

Table of n, a(n) for n=1..36.

Art of Problem Solving, Problem A112

Peter Vandendriessche and Hojoo Lee, Problems in Elementary Number Theory (see problem A112, p. 15). [Via Wayback Machine]

Formula

Conjectures from Chai Wah Wu, Jan 28 2024: (Start)

a(n) = 3*a(n-2) - a(n-4) for n > 4.

G.f.: x*(-4*x^3 - x^2 + 11*x + 4)/(x^4 - 3*x^2 + 1). (End)

Maple

nn:=10^6:a:=4:
for b from a+1 to nn do:
x:=(a^2-5)/b:y:=(b^2-5)/a:
if x>0 and y>0 and gcd(a, b)=1 and x=floor(x) and y=floor(y)
then
printf(`%d, `, a): printf(`%d, `, b):a:=b:
else fi:
od:

Mathematica

nn = 10^6; a = 4; Reap[For[b = a+1, b <= nn, b++, x = (a^2-5)/b; y = (b^2-5)/a; If[x>0 && y>0 && GCD[a, b] == 1 && x == Floor[x] && y == Floor[y], Print[a, " ", b]; Sow[a]; Sow[b]; a = b]]][[2, 1]] (* adapted from Maple *)
(* Second program: *)
Clear[a]; a[n_] := 2^(-n-2)*((7-3*Sqrt[5])*(1-Sqrt[5])^n-(-Sqrt[5]-1)^(n+1) - (Sqrt[5]-1)^(n+1) + (3*Sqrt[5]+7)*(Sqrt[5]+1)^n); Table[a[n] // Simplify, {n, 1, 36}] (* Jean-François Alcover, Jan 25 2017 *)

Crossrefs

Cf. A001622, A002878, A104457.

Sequence in context: A210693 A168212 A014449 * A355608 A098060 A268232

Adjacent sequences: A281384 A281385 A281386 * A281388 A281389 A281390

Keyword

nonn

Author

Michel Lagneau, Jan 21 2017

Status

approved