- (x1^2 + y1^2) (x2^2 + y2^2) = (x1 x2 - y1 y2)^2 + (x1 y2 + x2 y1)^2 =

(x1 x2 + y1 y2)^2 + (x1 y2 - x2 y1)^2

expresses the sum of two squares factorization relationships.

Suppose we wish to look at the special subset of form { z such that z =

(t^2 + 1) = p q, where t is integer, and p and q are primes.}

We can identify elements of that subset by:

Pick x1,x2,y1,y2 such that x1 x2 - y1 y2 = 1 and

x1^2 + y1^2 is prime

and

x2^2 + y2^2 is prime.

Is it possible to do this?

Kermit - On Sat, Mar 10, 2012 at 12:10 PM, Kermit Rose <kermit@...> wrote:
> Suppose we wish to look at the special subset of form { z such that z =

depending on the size of the numbers,

> (t^2 + 1) = p q, where t is integer, and p and q are primes.}

>

> We can identify elements of that subset by:

>

> Pick x1,x2,y1,y2 such that x1 x2 - y1 y2 = 1 and

> x1^2 + y1^2 is prime and x2^2 + y2^2 is prime.

>

> Is it possible to do this?

I think it's faster to consider products of primes and check whether

pq-1 is a square.

Maximilian - On Sat, Mar 10, 2012 at 1:31 PM, Maximilian Hasler

<maximilian.hasler@...> wrote:> On Sat, Mar 10, 2012 at 12:10 PM, Kermit Rose <kermit@...> wrote:

forprime(p=1,999,forprime(q=1,p,issquare(p*q-1)&print1(p*q",")))

>> Suppose we wish to look at the special subset of form { z such that z =

>> (t^2 + 1) = p q, where t is integer, and p and q are primes.}

>

> depending on the size of the numbers,

> I think it's faster to consider products of primes and check whether

> pq-1 is a square.

10,26,65,145,82,901,122,2501,2117,1157,485,5777,226,10001,1937,785,6401,362,20737,4097,3601,626,12997,18497,1765,10817,75077,111557,70757,842,64517,2305,81797,2705,7397,266257,1226,37637,23717,254017,11237,3365,320357,9217,144401,448901,1522,3845,207937,276677,60517,270401,527077,244037,104977,38417,712337,698897,

This (in increasing order) is oeis.org/A144255 : semiprimes of the form n^2+1

Maximilian - On 3/11/2012 9:03 AM, primenumbers@yahoogroups.com wrote:
> 1c. Re: seeking numerical example

Given that 10 = 3^2 + 1,

> Posted by: "Maximilian Hasler"maximilian.hasler@... maximilian_hasler

> Date: Sat Mar 10, 2012 9:37 am ((PST))

>

> On Sat, Mar 10, 2012 at 1:31 PM, Maximilian Hasler

> <maximilian.hasler@...> wrote:

>> > On Sat, Mar 10, 2012 at 12:10 PM, Kermit Rose<kermit@...> wrote:

>>> >> Suppose we wish to look at the special subset of form { z such that z =

>>> >> (t^2 + 1) = p q, where t is integer, and p and q are primes.}

>> >

>> > depending on the size of the numbers,

>> > I think it's faster to consider products of primes and check whether

>> > pq-1 is a square.

> forprime(p=1,999,forprime(q=1,p,issquare(p*q-1)&print1(p*q",")))

> 10,26,65,145,82,901,122,2501,2117,1157,485,5777,226,10001,1937,785,6401,362,20737,4097,3601,626,12997,18497,1765,10817,75077,111557,70757,842,64517,2305,81797,2705,7397,266257,1226,37637,23717,254017,11237,3365,320357,9217,144401,448901,1522,3845,207937,276677,60517,270401,527077,244037,104977,38417,712337,698897,

>

> This (in increasing order) is oeis.org/A144255 : semiprimes of the form n^2+1

>

> Maximilian

I require that

x1 x2 - y1 y2 = 1

and

x1 y2 + x2 y1 = 3

and

x1^2 + y1^2 = 5

and

x2^2 + y2^2 = 2

This gives possible solution

x1 = 2, y1 = 1, x2 = 1, y2 = 1

Ok. Thank you Max.

It appears that I have incompletely analyzed the requirements.

I might or might not get back to you on this problem depending on

my success in more completely analyzing the requirements of it.

Kermit