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Contradict Goldbach (Posted on 2015-03-03) Difficulty: 3 of 5
In 1752 Goldbach conjectured that every odd integer can be written in the form p + 2a^2, where p is unity or a prime and a is allowed to be zero.
This conjecture is found to be false - there are only two known composite exceptions, 5777 (=53*109) being one of them.

Please find the other, it is below 99,999.

No Solution Yet Submitted by Ady TZIDON    
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Solution computer assisted solution Comment 2 of 2 |
5777  (given)
5993  (the other one)

from

DefDbl A-Z
Dim wd(10) As String, w As String, sz, crlf$


Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
  For n = 3 To 99999 Step 2
   If prmdiv(n) < n Then
        p = 1: found = 0
        Do
          q = (n - p) / 2
          If q = Int(q) Then
            a = Int(Sqr(q) + 0.5)
            If a * a = q Then
              found = 1: Exit Do
            End If
          End If
          p = nxtprm(p)
        Loop Until p > n
        If found = 0 Then Text1.Text = Text1.Text & n & crlf
   End If
  Next

End Sub

Function prm(i)
  Dim p As Long
  Open "17-bit primes.bin" For Random As #111 Len = 4
  Get #111, i, p
  prm = p
  Close 111
End Function

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
    n = n + 1
  Wend
  nxtprm = n
End Function


  Posted by Charlie on 2015-03-03 13:24:17
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