[seqfan] Re: A169580 and Catalan's empirical claim that the triple of any odd square not divisible by 5 is a sum of squares of three primes other than 2 and 3.

[Richard Guy]

No doubt Catalan, along with Goldbach, D N Lehmer
and many others a hundred or more years ago,
regarded  1  as a prime, and in many contexts
it's useful to take 1 as the zeroth prime.
I believe that that is needed here:

3 = 1^2 + 1^2 + 1^2

27 = 5^2 + 1^2 + 1^2

147 = 11^2 + 5^2 + 1^2 = 7^2 + 7^2 + 7^2

243 = 13^2 + 7^2 + 5^2 = 11^2 + 11^2 + 1^2

363 = 19^2 + 1^2 + 1^2 = 17^2 + 7^2 + 5^2 =
     = 13^2 + 13^2 + 5^2 = 11^2 + 11^2 + 11^2

Presumably this is one of those obviously
true but quite unprovable theorems?  Any
comments on  number of representations ??
Can one manage without 1 from here on ??   R.

On Wed, 3 Mar 2010, Jonathan Post wrote:

> Apologies; I hit "send" before adding:
>
> Catalan stated and Realis proved that every power of 3 is a sum of
> three squares relatively prime to 3.
>
> On Wed, Mar 3, 2010 at 9:58 AM, Jonathan Post <jvospost3 at gmail.com> wrote:
>> Catalan stated empirically that the triple of any odd square not
>> divisible by 5 is a sum of squares of three primes other than 2 and 3.
>>
>> Is this worth adding as a comment about a subsequence of to njas's new seq
>> A169580 Squares of the form x^2+y^2+z^2 with x,y,z positive integers.
>>_______________________________________________
>
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