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From: Ludovicus on 30 Jul 2010 12:55
On 29 jul, 11:17, hagman <goo...(a)von-eitzen.de> wrote:
> On 18 Jul., 14:27, Ludovicus <luir...(a)yahoo.com> wrote:
>
> > Thanks to Tim Little for the answer to my former question on
> > Fibonacci.
> > I found its demonstration  in Vorobiev's "Numeros de Fibonacci"
> > Editorial Mir  Moscow 1974
>

> > "Every odd number can be decomposed in the sum of two numbers
> > such that the sum of their squares is a prime number."
> > Example: 33 =  5 + 29   ;    5^2 + 28^2 = 809  (prime)
> > Ludovicus
>
> It's at least somewhat likely:
> Between n and 2 n  there are about n/(ln(2n)) primes p = 1 mod 4.
> Each of these can be uniquely written as p = a^2 + b^2 with a even, b
> odd.
> This gives rise to a decomposition k = a + b for some odd number
> sqrt(n) < k < 2 sqrt(n).
> Naively, one would expect each k in that range to be hit about  2
> sqrt(n)/ln(2n)  times

No. Between n and 2n there are about .5n / ln(2n) primes p=1 mod 4.
Your formula do not give the true value of the number of primes
x^2 + y^2 for a given odd number. That number is very irregular
and depends of the odd number last digit.

This is a table from computed experiments with odds ended in 1.

ODD NUMBER CASES x+y PRIMES PRIMES / CASES

331 165 42 0.2545

601 300 62 0.2067

1301 650 129 0.1985

3601 1800 343 0.1906

6001 3000 540 0.1800

10001 5000 785 0.1570

42001 21000 2754 0.1311

100001 50000 5532 0.1106

Ludovicus
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