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?
Give reasons for your answer.
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$
Private Sub Form_Load()
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
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Posted by Charlie
on 2015-12-31 10:18:36 |
It seems...
