## Re: 24n+1 is prime

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• As mentioned by others, you can prove x*y == 0 mod 3 but I am having a hard time connecting the relevance of this to the original question, uniquely. Adam
Message 1 of 6 , Mar 28, 2003
As mentioned by others, you can prove x*y == 0 mod 3 but I am having
a hard time connecting the relevance of this to the original
question, "uniquely."

--- In primenumbers@yahoogroups.com, "Jon Perry" <perry@g...> wrote:
> Can someone help me prove this?
>
> If 24n+1 is prime, and as this is a prime 1mod4, then 24n+1 = x^2 +
y^2
> uniquely.
>
> {
> for(n=1,1000,p=24*n+1;if (n%100==0,print(n));
> if (isprime(p),sp=sqrt(p);
> for (i=1,sp,for (j=i,sp,k=i*i+j*j;
> if (k==p,ok=0;
> if (i%3==0,ok=1);
> if (j%3==0,ok=1);
> if (ok==0,print(p": "i", "j" -> "ok)))))))
> }
>
> This code confirms that for n to 1000, one of x or y is ALWAYS
divisible by
> 3.
>
> Jon Perry
> perry@g...
> http://www.users.globalnet.co.uk/~perry/maths/
> BrainBench MVP for HTML and JavaScript
> http://www.brainbench.com
• Fermat showed that every prime of the form 4n+1 can be expressed as the sum of two squares in exactly one way. As a prime of the form 24n+1 is 1 mod 3, x^2+y^2
Message 2 of 6 , Mar 28, 2003
Fermat showed that every prime of the form 4n+1 can be expressed as the sum of
two squares in exactly one way.

As a prime of the form 24n+1 is 1 mod 3, x^2+y^2 must be 1 mod 3, and that can
happen if exactly one of them is divisible by 3.

What is the problem here?

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• Actually, Euler proved it, Fermat conjectured it: http://mathworld.wolfram.com/Fermat4nPlus1Theorem.html Jon Perry perry@globalnet.co.uk
Message 3 of 6 , Mar 28, 2003
Actually, Euler proved it, Fermat conjectured it:

http://mathworld.wolfram.com/Fermat4nPlus1Theorem.html

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Paul Jobling [mailto:Paul.Jobling@...]
Sent: 28 March 2003 15:09
Subject: RE: [PrimeNumbers] Re: 24n+1 is prime

Fermat showed that every prime of the form 4n+1 can be expressed as the sum
of
two squares in exactly one way.

As a prime of the form 24n+1 is 1 mod 3, x^2+y^2 must be 1 mod 3, and that
can
happen if exactly one of them is divisible by 3.

What is the problem here?

__________________________________________________
Virus checked by MessageLabs Virus Control Centre.

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