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Comb(k,2) (Posted on 2018-03-29) Difficulty: 3 of 5
For which k ≥ 3 is k(k-1)/2 (i.e. k choose 2) one more than a power of a prime number?

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts solutions through k=1,000,000 | Comment 2 of 4 |
k  C(k,2)
3    3     2^1+1
4    6     5^1+1
5   10     3^2+1
8   28     3^3+1

from

DefDbl A-Z
Dim crlf$, fct(20, 1)

Private Sub Form_Load()
 Form1.Visible = True
 Text1.Text = ""
 crlf = Chr(13) & Chr(10)
 
 For k = 3 To 1000000
   v = k * (k - 1) / 2
   f = factor(v - 1)
   If f = 1 Then
     Text1.Text = Text1.Text & k & Str(v) & crlf
   End If
 Next

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

Function factor(num)
 diffCt = 0: good = 1
 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
   If INKEY$ = Chr$(27) Then s$ = Chr$(27): Exit Function
 Loop
 If n > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = n: fct(diffCt, 1) = 1
 factor = diffCt
 Exit Function

DivideIt:
 cnt = 0
 Do
  q = Int(n / dv)
  If q * dv = n And n > 0 Then
    n = q: cnt = cnt + 1: If n > 0 Then limit = Sqr(n) Else limit = 0
    If limit <> Int(limit) Then limit = Int(limit + 1)
   Else
    Exit Do
  End If
 Loop
 If cnt > 0 Then
   diffCt = diffCt + 1
   fct(diffCt, 0) = dv
   fct(diffCt, 1) = cnt
 End If
 Return
End Function

I could have speeded up runtime if I had modified the factor function to stop after finding a second prime factor, but that would have required extra coding time.


  Posted by Charlie on 2018-03-29 10:11:46
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