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4 primes (Posted on 2015-05-06) Difficulty: 3 of 5
There is a prime quadruple i.e. four consecutive primes such that:

a. Each one of them consists of distinct digits.
b. For each one of them the sum of the digits' cubes is a prime number.
c. In the new sequence of 4 primes none has repeated digits.

I believe there is a poor chance of multiple solutions but you are welcome to explore the issue after finding "my" quadruple (all are 3 digit primes).

See The Solution Submitted by Ady TZIDON    
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Solution computer solution | Comment 2 of 3 |
Each row of four successive primes is followed by a line with their sum of digits' cubes:

821 823 827 829
521 547 863 1249

2063 2069 2081 2083
251 953 521 547

2069 2081 2083 2087
953 521 547 863

2081 2083 2087 2089
521 547 863 1249

850613 850631 850637 850673
881 881 1223 1223

The four-digit cases are really one set of six successive primes:

2063 2069 2081 2083 2087 2089
251  953  521  547  863  1249


The six-digit case does not fit the puzzle's requirement that there be no repeat digit within each of the digit cube totals.


DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True

 Do
   pppp = ppp
   ppp = pp
   pp = p
   p = nxtprm(p)
   sc = sumcubes(p)
   If sc > 0 And prmdiv(sc) = sc Then
      succCt = succCt + 1
      If succCt > 3 Then
        Text1.Text = Text1.Text & pppp & Str(ppp) & Str(pp) & Str(p) & crlf
        Text1.Text = Text1.Text & sumcubes(pppp) & Str(sumcubes(ppp)) & Str(sumcubes(pp)) & Str(sumcubes(p)) & crlf
        Text1.Text = Text1.Text & crlf
        DoEvents
      End If
    Else
      succCt = 0
   End If
 Loop Until p > 1000000
 Text1.Text = Text1.Text & "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
    n = n + 1
  Wend
  nxtprm = n
End Function

Function sumcubes(x)
 s$ = LTrim(Str(x))
 tot = 0
 For i = 1 To Len(s)
   If InStr(Mid(s, i + 1), Mid(s, i, 1)) > 0 Then sumcubes = 0: Exit Function
   v = Val(Mid(s, i, 1))
   tot = tot + v * v * v
 Next
 sumcubes = tot
End Function


  Posted by Charlie on 2015-05-06 15:40:46
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