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Re: What squares are the sums of two or more cubes?
Posted:
Oct 19, 2012 9:18 PM
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On Fri, 19 Oct 2012 02:46:53 -0700, Pubkeybreaker wrote:
> On Oct 18, 9:57 pm, James Waldby <n...@valid.invalid> wrote:
>> On Fri, 19 Oct 2012 01:06:32 +0100, Peter Percival wrote:
>> > What squares of positive integers are the sums of two or more distinct
>> > cubes of positive integers?
>>
>> > 9 = 1 + 8, 36 = 1 + 8 + 27, 100 = 1 + 8 + 27 + 64, 225 = 1 + 8 + 27 + 64 +
>> > 125, etc.
>>
>> > On the LHSs are the squares of the triangular numbers. Are there any
>> > examples not in that sequence?
...
>> Let S be the set of sums of three cubes of numbers that are in the
>> range 0 to n. There are n^3 such sums. Even though many of the sums
>> have duplicated values, S nevertheless covers on the order of n^3
>> values in the range from 0 to 3 n^3, so naively the chance of a
>> number being the sum of three cubes is in the neighborhood of 1/3.
>> Let T be the set of sums of four cubes of numbers that are in the
>> range 0 to n. This gives some fraction of n^4 distinct sums for
>> numbers in the range 0 to 4 n^3, so on the average there are in the
>> neighborhood of n/4 ways to write a number less than 4 n^3 as the
>> sum of 4 cubes. However, among even small squares it looks like
>> 14^2, 18^2, 20^2, and 26^2 require five cubes in their sums:
>...
> The OP asked for DISTINCT cubes; Waring's problem is not directly
> applicable.
True, so 17 of the 18 sums I mentioned (and have snipped) were
irrelevant and the links for limits with 9 or 7 cubes (also snipped)
are irrelevant. But results of Nicomachus and Liouville are for sets
of distinct cubes, and the order-of-magnitude estimates given above
ought to apply if n is not too small. Only 12 numbers less than 50 can
be represented as the sums of 4 or 5 distinct positive cubes, but 49 of
the numbers less than 100 can be, with growing numbers of different
ways to represent each. For example, 99^2 is the sum of 4 or 5 distinct
positive cubes 8 ways: (1, 3, 8, 21), (8, 9, 10, 12, 18),
(1, 2, 4, 12, 20), (1, 8, 10, 15, 17), (2, 4, 5, 7, 21),
(2, 4, 9, 10, 20), (2, 5, 13, 15, 16), (4, 6, 8, 16, 17).
--
jiw
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