## Tuesday, April 07, 2015

### Two cutoffs about Warings problem for cubes

Known:
1. All numbers except 23 can be written as the sum of 8 cubes
2. All but a finite number of numbers can be written as the sum of 7 cubes
3. There are an infinite number of numbers that cannot be written as the sum of 3 cubes(this you can prove yourself, the other two are hard, deep theorems).
Open: Find x such that:
1. All but a finite number of numbers can be written as the sum of x cubes.
2. There exists an infinite number of numbers that cannot be written as the sum of x-1 cubes.
It is known that 4 ≤ x ≤ 7

Lets say you didn't know any of this and were looking at empirical data.

1. If you find that every number ≤ 10 can be written as the sum of 7 cubes this is NOT interesting because 10 is too small.
2.  If you find that every number ≤ 1,000,000 except 23 can be written as the sum of 8 cubes this IS interesting since 1,000,000 is big enough that one thinks this is telling us something (though we could be wrong). What if you find all but 10 numbers (I do not know if that is true) ≤ 1,000,000 are the sum of seven cubes?
Open but too informal to be a real question: Find x such that
1. Information about sums-of-cubes for all numbers ≤ x-1 is NOT interesting
2. Information about sums-of-cubes for all numbers ≤ x IS interesting.
By the intermediate value theorem such an x exists. But of course this is silly. The fallacy probably relies on the informal notion interesting'. But a serious question: How big does x have to be before data about this would be considered interesting? (NO- I won't come back with what about x-1').

More advanced form: Find a function f(x,y) and constants c1 and c2 such that
1. If f(x,y) ≥ c1  then the statement all but y numbers ≤ x are the sum of 7 cubes is interesting.
2. If f(x,y) ≤  c2 then the statement all but y numbers ≤ x are the sum of 7 cubes is not interesting.
To end with a more concrete question: Show that there are an infinite number of numbers that cannot be written as the sum of 14 4th powers.