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A rare palindrome (Posted on 2017-03-18) Difficulty: 4 of 5
Please consider the following equations:

64446 = 32213 + 32233
&
64446 = 33223 + 31223

This is the smallest example of a palindromic number S that is a sum of two consecutive primes and on the other hand equals the sum of the reversals of those two primes (also primes, albeit not consecutive).

Please provide another sample(s) with such peculiar feature.

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution Comment 1 of 1
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()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 
 p = 2
 Do
   newp = nxtprm(p)
   tot = p + newp
   st$ = LTrim(Str(tot))
   good = 1
   For i = 1 To Len(st) / 2
     If Mid(st, i, 1) <> Mid(st, Len(st) + 1 - i, 1) Then good = 0: Exit For
   Next
   If good Then
    st1$ = LTrim(Str(p)): st2$ = LTrim(Str(newp))
    st1r$ = ""
    For i = 1 To Len(st1)
      st1r$ = Mid(st1, i, 1) + st1r
    Next
    st2r$ = ""
    For i = 1 To Len(st2)
      st2r$ = Mid(st2, i, 1) + st2r
    Next
    r1 = Val(st1r): r2 = Val(st2r)
    If r1 + r2 = tot Then
      If prmdiv(r1) = r1 And prmdiv(r2) = r2 Then
        Text1.Text = Text1.Text & p & Str(newp) & "    " & tot & Str(r1) & Str(r2) & crlf
      End If
    End If
   End If
   DoEvents
   oldp = p: p = newp
   If p = 32213 Then
    xx = xx
   End If
 Loop


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 Or n < 2
    n = n + 1
  Wend
  nxtprm = n
End Function

shows the two successive primes and their palindromic total, followed by the reversed values that are also prime.

The first two below are trivial 1-digit primes with a 1-digit "palindrome" each. The next is the case given in the puzzle, and then comes the sought answer,

132,040,201 + 132,040,261 = 264,080,462 with the reverse primes being 102,040,231 and 162,040,231.

2 3    5 2 3
3 5    8 3 5
32213 32233    64446 31223 33223
132040201 132040261    264080462 102040231 162040231

The program was manually terminated at a point when the lower prime being checked was 1,387,665,827.


  Posted by Charlie on 2017-03-18 13:48:46
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