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A false conjecture (Posted on 2017-01-27) Difficulty: 4 of 5
"Every positive number bigger than 1 can be represented as a sum of a square, a nonnegative cube and two positive Fibonacci numbers".
Example: 113=100+0+5+8

NOT SO!

Find the smallest integer n justifying the title of this puzzle.

Rem: It is quite a big number!

See The Solution Submitted by Ady TZIDON    
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Some Thoughts computer attempt | Comment 1 of 7
The below program check numbers up to 200,000 and finds no counterexample up to that value:

DefDbl A-Z
Dim crlf$, fib(50)


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 fib(1) = 1: fib(2) = 2
 n = 3
 Do
   fib(n) = fib(n - 1) + fib(n - 2)
   n = n + 1
 Loop Until fib(n - 1) > 200000
 mxfib = n - 2
 Text1.Text = Text1.Text & mxfib & Str(fib(mxfib)) & crlf
 Open "false conjecture.bin" For Binary As #2
 
 one$ = "1"
 For s = 0 To 1414
   n = s * s
   For c = 0 To 125
    DoEvents
     n = n + c * c * c
     If n > 200000 Then n = n - c * c * c: Exit For
     For f1 = 1 To mxfib
      DoEvents
       n = n + fib(f1)
       If n > 200000 Then n = n - fib(f1): Exit For
       For f2 = f1 To mxfib
        DoEvents
         n = n + fib(f2)
         If n <= 200000 Then
           Put #2, n, one
         End If
         n = n - fib(f2)
       Next f2
       n = n - fib(f1)
     Next f1
     n = n - c * c * c
   Next c
 Next
 
 d$ = " "
 For n = 1 To 200000
   Get #2, n, d$
   If d$ <> "1" Then
     Text1.Text = Text1.Text & n & crlf
   End If
   DoEvents
 Next n
 
 Close 2
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

Edited on January 27, 2017, 12:38 pm
  Posted by Charlie on 2017-01-27 12:37:47

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