[seqfan] Re: 448 ways

[Joseph Myers]

Jacobi showed that the number of representations of n as a sum of four 
squares, where the order and signs of the numbers squared matter, is 8 
times the sum of the divisors of n that are not multiples of 4.  So for n 
odd this is at least 8n, and when the order and signs do not matter it is 
at least n/48.

So for sums of five squares, look at n-k^2 where k has opposite parity to 
n, and each representation may appear at most five times with different 
choices of k.  Bounding this sum of (n-k^2)/240 by an integral, I get that 
the number of representations of n as the sum of five squares is at least 
(1/720)n^{3/2} - n/480 + 1/1440.

-- 
Joseph S. Myers
jsm at polyomino.org.uk

On Thu, 16 Nov 2017, Frank Adams-Watters via SeqFan wrote:

> A good average for A002635 (number of ways to represent n 
> as the sum of 4 squares) would enable us to improve this lower 
> bound considerably. This would also be interesting in and of itself.
> 
> (One has to look for an average, no a lower bound, as there are 
> infinitely many n with A002635(n) equal to any given k > 0.)
> 
> There is no such average in A002635 itself; I did not look at any 
> of the references.
> 
> Franklin T. Adams-Watters
> 
> 
> -----Original Message-----
> From: israel <israel at math.ubc.ca>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Wed, Nov 15, 2017 8:52 pm
> Subject: [seqfan] 448 ways
> 
> A295159 (recently submitted by Robert Price) is titled "Smallest number 
> with exactly n representations as a sum of five nonnegative squares." 
> 
> Of course this raises the question, does such a number always exist?
> It appears to me that although such numbers exist for n <= 447, there 
> may not be such a number in the case n=448. I have determined that 
> there is no such number <= 25000, and from looking at A000174 it 
> seems unlikely that any number x > 25000 would have A000174(x) 
> as small as 448. Can anybody prove my conjecture?One possible 
> route to proving the conjecture would involve good lower bounds for 
> A000174(x). One simple lower bound is floor(sqrt(x) - sqrt(x/2)), from 
> the following argument: if y is any integer in the interval (sqrt(x/2), sqrt(x)], 
> we can represent x as the sum of 5 squares of which one is y^2 
> (since x - y^2 is the sum of 4 squares), and since y^2 > x/2 the others 
> are all < y^2, making these all distinct representations. To make that 
> lower bound 448 would require x >= 2339578, and that's too big for 
> me to search. But maybe better lower bounds are known.Cheers,
> 
> Robert Israel--Seqfan Mailing list - http://list.seqfan.eu/
> 
> --
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