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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


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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re(2): computer attempt | Comment 4 of 7 |
(In reply to re: computer attempt by Charlie)

I have thought of an optimization to the search method.  I have it running currently on my computer.  I will let you know if it gives a result.

First, I generated a list of the first 1000 fibonacci numbers.  I then generated a list of all the distinct sums of two fibonacci numbers.  From this I generated intervals within which no number can be expressed as the sum of two fibonacci numbers.

Now, something to realize is that because we are allowing zero to be in the list of squares and cubes, then if a number N disproves the conjecture then it can not be represented as the sum of two fibonacci numbers.

Thus what my search is doing is the same thing that Charile's does, however I have restricted the search to the numbers in the intervals I found.

I do not have access to my workstation so I am running it right now on a slower laptop.  Perhaps you will have better luck using this method Charlie.

  Posted by Daniel on 2017-01-29 13:12:03
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