- I find this all very interesting, it is new to me. What specifically

caught my attention was the linear representation of primes generated

by the equation x^2 + p*y^2, where p is prime. The nature of the

linear representation is not always the same from prime to prime. Is

there are rule as Sleephound queried, I'm not sure.

Below are the representations for p=2, p=3, p=5, p=7 and p=11. The

p=11 result is courtesy of Sleephound's data, not Wolfram's site.

From the Wolfram site,

For p=2 :

All primes that can be expressed as x^2 + 2y^2 can also be

represented as 8n+1 or 8n+3. Conversely, if I understand it

correctly, all primes of the form 8n+1 or 8n+3 can be expressed as

x^2 + 2y^2.

All primes that can be expressed as x^2 - 2y^2 can also be

represented as 8n+1 or 8n+7.

Observation: There is partial intersection between the two groups.

One quarter of the primes are not represented at all, that is primes

of the form 8n + 5.

For p=3:

All primes that can be expressed as x^2 + 3y^2 can also be

represented as 6n+1.

All primes that can be expressed as x^2 - 3y^2 can also be

represented as 12n+1.

Observation: One group is entirely a subset of the other. One half of

the primes are not represented, that is primes of the form 6n-1.

For p=5:

All primes that can be expressed as x^2 + 5y^2 can be also be

represented as 20n+1 or 20n+9.

All primes that can be expressed as x^2 - 5y^2 can also be

represented as 10n+1 or 10n+9.

Observation: One group is entirely a subset of the other. One half of

the primes are not represented, that is primes of the form 10n+3 and

10n+7.

For p=7:

All primes that can be expressed as x^2 + 7y^2 can also be

represented as 14n+1 or 14n+9 or 14n+25.

All primes that can be expressed as x^2 - 7y^2 can also be

represented as 28n+1 or 28n+9 or 28n+25.

Observation: One group is entirely a subset of the other. One half of

the primes are not represented, that is primes of the form 14n+3 and

14n+5 and 14n+9.

For p=11:

All primes that can be expressed as x^2 + 11y^2 can also be

represented as 22n+1, 22n+3, 22n+5, 22n+9 or 22n + 15. But the

converse is not true; not all primes of the form 22n+1, 22n+3, 22n+9

or 22n+15 can be written as x^2 + 11y^2.

All primes that can be expressed as x^2 - 11y^2 can be also

represented as 22n+1 or 22n+3 or 22n+9 or 22n+15. But again, I don't

think the converse is true.

Observation: I'm not sure if the two groups overlap even though they

have the same linear representation. And the fact that the converse

is not true in this cases sets it apart from the earlier primes. I

wonder if all the rest of the primes after 11 (13,17,19 ...) are like

11 in this regard?

Final Note:

Also, I noticed from Wolfram's table that all primes of the form x^2

+ y^2 can be written as 4n+1, and (I am supposing) the converse is

true, all primes of the form 4n+1 can be written as x^2 + y^2. That

is half the primes. And as we know all of the primes can be written

as x^2 - y^2.

It seems that no expression x^2 - by^2 combined with x^2 + by^2 will

yield all the primes, except of course when b = 1. But it appears

that the combined *three* expressions, x^2 + y^2, x^2 + 2y^2 and x^2 -

2y^2 will yield all the primes since they cover all the

possibilities: 8n+1, 8n+3, 8n+5 and 8n+7. I doubt that any other

combination of three will yield all the primes. Wait, could it be

that no combination of four will yield all the primes, even if up to

two of the above three expressions are used? Further, no combination

of 5, or 6, or 7 or ....?

Mark

--- In primenumbers@yahoogroups.com, "sleephound" <sleephound@y...>

wrote:> Primes of the form X^2+Y^2 are of the form 4n+1, Primes of the form

> X^2+2Y^2 are of the form 8n+1 or 8n+1. Some other examples of

> quadratic forms yielding primes of linear form are listed here:

>

> http://mathworld.wolfram.com/PrimeRepresentation.html

>

> Primes of the form X^2+11Y^2 begin 47, 53, 103, 163, 199. There

> doesn't appear to be a linear form that separates these primes into

> two groups - I've checked through 4000. Is there some simple rule

> that characterizes these primes? Is there a simple rule for when a

> linear characterization exists? - In a message dated 08/07/03 02:14:47 GMT Daylight Time,

tomabeti@... writes:

>> Can you reduce it? (I'm no pari expert) - it ought to be equivalent to my

Thank you.

>> cubic.

>

> x^3 - x^2 + x + 1

Meanwhile, (pari Grand Master) David Broadhursthas emailed me with> henri cohen's "minimal height" form for your d=-44 cubic is

So that's good ! - there's complete agreement between your theoretical

> x^3-x^2+x+1

Hilbert polynomial and my experimentally discovered cubic.

Mike Oakes

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