Germain Primes and the Homogenous Integer Function Q(x,y)

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In summary, the equation Q(x,y) = (x^2 + xy + y^2)^((p-3)/2) can be solved for x and y, provided p-3 is known.
  • #1
vantheman
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Reference: www.mathpages.com/home367.htm[/URL]
On page 2 of reference the formula is given
(x+y)^p - x^p - y^p = pxy(x=y)Q(x,y) where Q(x,y) is a homogenous integer function of degree p-3.
If we insert a number of different value of p into the equation, it appears that
Q(x,y) = (x^2 = xy + y^2)^((p-3)/2)

Is there an easy way to prove this without getting lost in infinite series calculations, or is there a proof already in print?
 
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  • #2
You seem in a much better position to investigate this than anybody else, particularly since your reference can not be found.

(x+y)^p - x^p - y^p = pxy(x=y)Q(x,y) where Q(x,y) is a homogenous integer function of degree p-3.

Are you sure you mean to write x=y? If so what is the point of Q(x,y)? Assuming you don't mean x=y, the Q(x,y) appears to be of degree p-2. Because we subtracted those of degree p, and pulled out a pxy from Q(x,y,) that leaves degree p-2.
 
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  • #3
Sorry for the errors. The correct web address is
www.mathpages.com/home/kmath367.htm[/URL]

The equal sign in the formula should have been "+" not "=". I neglected to hit the shift key.
I think p-3 is correct. When you substact x^p and y^p from the expansion, the results have a factor of pxy, so the x^p in now at the p-2 level. However, the remaining equation is divisible by (x+y), and this brings it to the p-3 level.

I'm an amateur looking for professional help. Is there an easy way to prove that

Q(x,y) = (x^2 + xy + y^2)^((n-3)/2) ?
 
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  • #4
I think you are assuming too much from other people here. It does help to state the definition of Q(x,y)

[tex]Q(x,y) = \frac{(x+y)^P-(x^P+y^P)}{(xy)(x+y)p}[/tex]

I thought it enough of a problem, letting p-1 = u, to show that p divides all terms: [tex](x+y)^u -\frac{x^u+y^u}{x+y}[/tex]
This can be found by induction on k: [tex]\frac{(p-1)!}{k!(p-1-k)!} \equiv (-1)^k Mod p [/tex]

So you maybe looking at an induction problem on n.
 
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  • #5
Thank you. It's very much appreciated.
 

1. What are Germain primes?

Germain primes are prime numbers that are also a safe prime, meaning they are of the form 2p+1 where p is also a prime number. They are named after French mathematician and physicist Pierre de Fermat's friend and correspondent, French mathematician and priest Marin Mersenne.

2. How are Germain primes related to safe primes?

Germain primes are a subset of safe primes, meaning all Germain primes are also safe primes. However, not all safe primes are Germain primes.

3. Are there infinitely many Germain primes?

It is currently unknown if there are infinitely many Germain primes, but it is widely believed to be true. As of 2021, only 2,423 Germain primes have been discovered, with the largest known Germain prime being 4,611,207 digits long.

4. How are Germain primes used in cryptography?

Germain primes play a crucial role in the Diffie-Hellman key exchange algorithm, which is widely used in cryptography to securely exchange secret keys over a public channel. The security of this algorithm relies on the difficulty of factoring large numbers, which includes Germain primes.

5. What is the significance of Germain primes in mathematics?

Germain primes have been studied extensively in number theory and have connections to various other mathematical concepts, such as Mersenne primes, Sophie Germain primes, and perfect numbers. They also have practical applications in cryptography and coding theory.

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