Pythagorean Primes and Gaussian Primes, divisibility question

In summary, the conversation discusses a problem involving a prime number that is the sum of squares of two integers and is also a norm of a Gaussian prime. The solution to the problem involves using Wilson's theorem and the factorial sign. The conversation also mentions that the problem is in a section on principal ideal domains and unique factorization domains, and that Wilson's theorem is often introduced in exercises around PIDs in abstract algebra books.
  • #1
joeblow
71
0
Here is an interesting problem that I've been thinking about for a while:

Let p be a prime s.t. p = 4m+1 for some integer m. Show that p divides n^2 + 1, where n = (2m)!

It comes from a section on principal ideal domains and unique factorization domains.

It is well-known that p is the sum of squares of two integers and thus is a norm of a Gaussian prime, and n^2 + 1 = (n+i)*(n-i).

However, I am not sure that this helps anything. Does anyone have any ideas. Your help is greatly appreciated.
 
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  • #2
I figured it out. For those of you who want a hint, try to use Wilson's theorem. (I have no idea why this was in the section on PID's.) When I saw the factorial sign, I immediately tried Wilson's theorem, but I needed to be much more clever in order to get it to work.
 
  • #3
In case other people find this thread I will post a solution:
[tex]2m+i \equiv (4m+1)-(2m+i) = 2m + 1 - i \pmod {4m+1}[/tex]
Thus,
[tex]\prod_{i=1}^{2m} (2m+i) \equiv \prod_{i=1}^{2m} (2m + 1 - i) = \prod_{i=1}^{2m}(2m+1-(2m+1-i)) = \prod_{i=1}^{2m} i = (2m)! \pmod {4m+1}[/tex]
This gives,
[tex](4m)! = (2m)!\prod_{i=1}^{2m} (2m+i) \equiv (2m)!^2 \pmod {4m+1}[/tex]
Now since 4m+1 is prime we can apply Wilson's theorem.
[tex]-1 \equiv (4m+1-1)! \equiv (2m)!^2 \pmod {4m+1}[/tex]
Thus [itex]4m+1 | (2m)!^2 + 1[/itex] which was to be shown.

I have no idea why this was in the section on PID's.
Wilson's theorem is introduced in the exercises around PIDs in many abstract algebra books. Usually this means that exercises on applying Wilson's theorem is scattered in nearby sections.
 
  • #4
Wilson's theorem is not presented anywhere near this problem. I would assume that this problem is supposed to help you determine which gaussian integers are irreducible.
 
  • #5


I am familiar with the concepts of Pythagorean Primes and Gaussian Primes, as well as divisibility and prime factorization. This problem is certainly an interesting one, and it highlights the connections between these different mathematical concepts.

To start, let's recall the definition of a Pythagorean Prime. A prime number p is considered a Pythagorean Prime if it can be expressed as the sum of two squares, p = a^2 + b^2. This is related to the Pythagorean Theorem, where the hypotenuse of a right triangle can also be expressed as the sum of two squares.

Next, we have Gaussian Primes, which are prime numbers in the complex plane. These are numbers of the form a+bi, where a and b are integers, and i is the imaginary unit. Similar to Pythagorean Primes, Gaussian Primes can also be expressed as the sum of two squares, p = a^2 + b^2.

Now, let's look at the problem at hand. We are given a prime number p = 4m+1 and we need to show that it divides n^2 + 1, where n = (2m)!. From the given information, we know that p can be expressed as the sum of two squares, p = (2m)^2 + 1^2. This means that p is a Pythagorean Prime.

Furthermore, we can also express n^2 + 1 as (n+i)*(n-i), where i is the imaginary unit. This means that n^2 + 1 is the product of two Gaussian Primes, (2m+i)*(2m-i). Since p is a Pythagorean Prime, it can also be expressed as the product of two Gaussian Primes, (2m+i)*(2m-i).

Therefore, we can see that p is a common divisor of both n^2 + 1 and (2m+i)*(2m-i). This means that p must divide n^2 + 1, as required.

In conclusion, we have shown that for a prime number p = 4m+1, where m is an integer, p will always divide n^2 + 1, where n = (2m)!. This is a result of the connections between Pythagorean Primes and Gaussian Primes, as well as the product representation of n^2 + 1.
 

1. What are Pythagorean Primes and Gaussian Primes?

Pythagorean Primes are positive integers that can be expressed as the sum of two squares, while Gaussian Primes are complex numbers that can only be divided by themselves and 1, and their conjugates. Both types of primes have unique properties and are of interest in number theory.

2. How are Pythagorean Primes and Gaussian Primes related?

While they are both types of primes, Pythagorean Primes and Gaussian Primes are not directly related. They are named after the mathematicians who studied them, Pythagoras and Carl Friedrich Gauss, respectively.

3. Is there a relationship between divisibility and Pythagorean Primes and Gaussian Primes?

Yes, divisibility plays a crucial role in determining whether a number is a Pythagorean Prime or a Gaussian Prime. For Pythagorean Primes, a number is divisible by a Pythagorean Prime if and only if it can be expressed as the sum of two squares. For Gaussian Primes, a number is divisible by a Gaussian Prime if and only if it can be factored into two smaller Gaussian Primes.

4. Are there any known divisibility rules for Pythagorean Primes and Gaussian Primes?

Yes, there are some known divisibility rules for Pythagorean Primes and Gaussian Primes. For example, all Pythagorean Primes greater than 5 have the form 4n+1, and all Gaussian Primes have the form 4n+3. These rules can help determine if a number is a Pythagorean Prime or a Gaussian Prime.

5. How are Pythagorean Primes and Gaussian Primes used in mathematics and science?

Both types of primes have applications in various fields of mathematics and science. For example, Pythagorean Primes are used in cryptography, while Gaussian Primes are used in number theory and signal processing. They also have connections to other areas of mathematics, such as Fermat's Last Theorem and the Riemann Hypothesis.

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