Define P(n) as the sum of all the positive integers that are less than n and relatively prime to n.
Find all possible positive integer values of n such that P(n) = 3n, and prove that there are no others.
DefDbl A-Z
Dim crlf$
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
Private Sub Form_Load()
ChDir "C:\Program Files (x86)\DevStudio\VB\projects\flooble"
Text1.Text = ""
crlf$ = Chr(13) + Chr(10)
Form1.Visible = True
DoEvents
For n = 1 To 1000000
p = phi(n)
If p = 3 * n Then
Text1.Text = Text1.Text & mform(n, "#####0") & mform(p, "#####0")
Text1.Text = Text1.Text & " "
For i = 1 To n
If gcd(i, n) = 1 Then Text1.Text = Text1.Text & mform(i, "#####0")
Next
Text1.Text = Text1.Text & crlf
End If
DoEvents
Next
End Sub
Function gcd(a, b)
x = a: y = b
Do
z = x Mod y
x = y: y = z
Loop Until z = 0
gcd = x
End Function
Function phi(n)
tot = 0
For i = 1 To n
If gcd(i, n) = 1 Then tot = tot + i
Next
phi = tot
End Function
finds
n p coprimes of n
7 21 1 2 3 4 5 6
9 27 1 2 4 5 7 8
14 42 1 3 5 9 11 13
18 54 1 5 7 11 13 17
After that the p's get larger and larger relative to 3n, but a proof that there are no others eludes me. I would think it would involve two cases: (i) the largest prime factor of n is no more than some given prime, such as 7; (ii) there is a prime factor larger than that prime. Either way we'd have to set a maximum limit on n.
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Posted by Charlie
on 2014-06-12 11:37:50 |
numbers without proof of completeness
