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 Infinite Illation Decision (Posted on 2015-12-31)
Each of X and Y is a positive integer, such that:
• X and Y are relatively prime, and:
• Each of (X2 – 5)/Y and (y2 – 5)/X is a positive integer.
Does there exist an infinite numbers of pairs (X,Y) satisfying the given conditions?

 No Solution Yet Submitted by K Sengupta Rating: 4.0000 (1 votes)

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 It seems... | Comment 1 of 3
It looks as if there's an infinite supply, with each lower value being the upper value from the preceding pair. Each upper value is approximately (3+sqrt(5))/2 times the lower value. That factor is phi + 1 where phi is the golden ratio.

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
141422324  370248451
370248451  969323029
969323029  2537720636
2537720636  6643838879
6643838879  17393796001

from

DefDbl A-Z
Dim crlf\$

Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

For tot = 6 To 5000
For x = 3 To tot / 2
DoEvents
y = tot - x
If gcd(x, y) = 1 Then
If (x * x - 5) Mod y = 0 Then
If (y * y - 5) Mod x = 0 Then
Text1.Text = Text1.Text & x & "  " & y & crlf
ysave = y
End If
End If
End If
Next
Next
Text1.Text = Text1.Text & crlf & crlf

x = ysave
rat = 1.5 + Sqr(5) / 2
Do
y = Int(x * rat) - 1
did = 0
Do
DoEvents
y = y + 1
If gcd(x, y) = 1 Then
q = Int((x * x - 5) / y)
r = (x * x - 5) - y * q
If r = 0 Then
q = Int((y * y - 5) / x)
r = (y * y - 5) - x * q
If r = 0 Then
Text1.Text = Text1.Text & x & "  " & y & crlf
did = 1
End If
End If
End If
Loop Until did
x = y
Loop Until y > 10000000000#

Text1.Text = Text1.Text & crlf & " done"

End Sub

Function gcd(a, b)
x = a: y = b
Do
q = Int(x / y)
z = x - q * y
x = y: y = z
Loop Until z = 0
gcd = x
End Function

I had first noticed the fact that the new low value was the old high value. After that, and placing the second part of the program, I noticed the ratio, after having first put in a guess of twice the low value and going up by 1 from there, as it's faster to start with the more accurate multiple.

To continue further without worrying about limits of precision, continuation is in UBASIC:

10           x = 3571
20           rat = 1.5 + Sqr(5) / 2
30           repeat
40             y = Int(x * rat) - 1
50             did = 0
60             repeat
80               y = y + 1
90               If gcd(x, y) = 1 Then
100                 q = Int((x * x - 5) / y)
110                 r = (x * x - 5) - y * q
120                 If r = 0 Then
130                 q = Int((y * y - 5) / x)
140                 r = (y * y - 5) - x * q
150                 If r = 0 Then
160                  :print  x ; "  " ; y
180                  :did = 1
190                 :End If
200                 :End If
210               :End If
220             Until did
230             x = y
240           Until y > 1000000000000000000000000

3571    9349
9349    24476
24476    64079
64079    167761
167761    439204
439204    1149851
1149851    3010349
3010349    7881196
7881196    20633239
20633239    54018521
54018521    141422324
141422324    370248451
370248451    969323029
969323029    2537720636
2537720636    6643838879
6643838879    17393796001
17393796001    45537549124
45537549124    119218851371
119218851371    312119004989
312119004989    817138163596
817138163596    2139295485799
2139295485799    5600748293801
5600748293801    14662949395604
14662949395604    38388099893011
38388099893011    100501350283429
100501350283429    263115950957276
263115950957276    688846502588399
688846502588399    1803423556807921
1803423556807921    4721424167835364
4721424167835364    12360848946698171
12360848946698171    32361122672259149
32361122672259149    84722519070079276
84722519070079276    221806434537978679
221806434537978679    580696784543856761
580696784543856761    1520283919093591604
1520283919093591604    3980154972736918051
3980154972736918051    10420180999117162549
10420180999117162549    27280388024614569596
27280388024614569596    71420983074726546239
71420983074726546239    186982561199565069121
186982561199565069121    489526700523968661124
489526700523968661124    1281597540372340914251
1281597540372340914251    3355265920593054081629
3355265920593054081629    8784200221406821330636
8784200221406821330636    22997334743627409910279
22997334743627409910279    60207804009475408400201
60207804009475408400201    157626077284798815290324
157626077284798815290324    412670427844921037470771
412670427844921037470771    1080385206249964297121989

 Posted by Charlie on 2015-12-31 10:18:36

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