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A281387 Pairs (x, y) of relatively prime positive integers such that (x^2 - 5)/y and (y^2 - 5)/x are both positive integers. 0
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 (list; graph; refs; listen; history; text; internal format)
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)). Lim_{n->inf} 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.

LINKS

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

Problems in Elementary Number Theory, Peter Vandendriessche, Hojoo Lee, July 11, 2007,Problem A112

Community of Problem Solving, Problem A112</a

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-Fran├žois Alcover, Jan 25 2017 *)

CROSSREFS

Cf. A001622, A104457.

Sequence in context: A210693 A168212 A014449 * A098060 A268232 A285633

Adjacent sequences:  A281384 A281385 A281386 * A281388 A281389 A281390

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jan 21 2017

STATUS

approved

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Last modified April 25 21:06 EDT 2019. Contains 322461 sequences. (Running on oeis4.)