I've found out that there is an infinite number of two successive integers such that neither can be expressed as the sum of 3 (or fewer) integer cubes.
To accept it as a fact you have either
- To trust me.............wrong choice
or
-To prove it............... Right!
Go for it!
All perfect cubes are either 0, +1, or -1 mod 9 depending on whether the number being cubed is 0, +1 or -1 mod 3.
In particular, any number that's 4 mod 9 cannot be the sum of three or fewer cubes - it takes at least four +1 cubes to reach that total. Similarly, any number that's 5 mod 9 (= -4 mod 9) can't be such a sum, as it takes at least four -1 cubes to reach that residue.
So in any set of integers between 9n and 9(n+1) there is a pair of successive integers (9n+4, 9n+5) that meet the criteria above.
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Posted by Paul
on 2020-06-19 16:18:12 |
trust no one