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 A false conjecture (Posted on 2017-01-27)
"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 No Rating

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 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)

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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