What is the probability of two randomly chosen integers being coprime?
Bonus(d4): How does the result relate fo the density
of square-free numbers?
Starting with a probability of 1, keep multiplying by 1 minus the square of the reciprocal of the next prime until the value levels off to an acceptable degree of accuracy. The program starts off with the probability reduced to the probability that the two numbers are not both even.
DefDbl A-Z
Dim crlf$
Private Sub Form_Load()
Text1.Text = ""
crlf$ = Chr(13) + Chr(10)
Form1.Visible = True
p = 3 / 4: prime = 2
Do
prev = p
prime = nxtprm(prime)
p = p * (1 - 1 / prime ^ 2)
If p = prev Then Exit Do
DoEvents
Loop Until prev - p < 0.00000000000001
Text1.Text = Text1.Text & p & Str(prev) & " done"
End Sub
Function prmdiv(num)
Dim n, dv, q
If num = 1 Then prmdiv = 1: Exit Function
n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0
If limit <> Int(limit) Then limit = Int(limit + 1)
dv = 2: GoSub DivideIt
dv = 3: GoSub DivideIt
dv = 5: GoSub DivideIt
dv = 7
Do Until dv > limit
GoSub DivideIt: dv = dv + 4 '11
GoSub DivideIt: dv = dv + 2 '13
GoSub DivideIt: dv = dv + 4 '17
GoSub DivideIt: dv = dv + 2 '19
GoSub DivideIt: dv = dv + 4 '23
GoSub DivideIt: dv = dv + 6 '29
GoSub DivideIt: dv = dv + 2 '31
GoSub DivideIt: dv = dv + 6 '37
Loop
If n > 1 Then prmdiv = n
Exit Function
DivideIt:
Do
q = Int(n / dv)
If q * dv = n And n > 0 Then
prmdiv = dv: Exit Function
Else
Exit Do
End If
Loop
Return
End Function
Function nxtprm(x)
Dim n
n = x + 1
While prmdiv(n) < n Or n < 2
n = n + 1
Wend
nxtprm = n
End Function
finds
0.607927106501875 .607927106501885
the first being the latest iteration and the second being the preceding iteration. Let's say 0.6079271065018 or 0.6079271065019 is a safe bet, or certainly 0.607927106502 with their implied degrees of precision.
Added:
BTW the program stopped at the 525815th prime, 7778509.
Edited on July 11, 2018, 11:43 am
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Posted by Charlie
on 2018-07-11 10:59:31 |
computer approximation
