Proving 3 as a Quadratic Non-Residue of Mersenne Primes | Number Theory Problem

In summary, a Mersenne prime is a prime number that can be expressed in the form 2^n - 1, where n is also a prime number. A quadratic non-residue of a prime number p is an integer x that is not a perfect square and when raised to the power of (p-1)/2 and modulo p, the remainder is not equal to 1. Proving 3 as a quadratic non-residue of Mersenne primes has important implications in number theory and cryptography as it can be used to strengthen security and has connections to other unsolved problems in mathematics. The proof for 3 as a quadratic non-residue involves using properties of quadratic residues and non-residues, as well as advanced
  • #1
ak_89
5
0
I was just working on some problems from a textbook I own (for fun).
I am not sure how to start this problem at all.

Here's the question: Show that 3 is a quadratic non-residue of all Mersenne primes greater than 3.

I honestly don't know how to start. If I could get some help to push me in the right direction that would be great.

Thanks
 
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  • #2
Do you know the law of quadratic reciprocity? If so, use it to work out examples for small Mersenne primes and look for some patterns.

Petek
 
  • #3
That helped me a lot. Thanks a bunch :wink:
 

1. What is a Mersenne prime?

A Mersenne prime is a prime number that can be expressed in the form 2n - 1, where n is also a prime number. They are named after the French mathematician Marin Mersenne who studied them in the 17th century.

2. What is a quadratic non-residue?

A quadratic non-residue of a prime number p is an integer x that is not a perfect square (i.e. there is no integer y such that x = y2) and when x is raised to the power of (p-1)/2 and modulo p, the remainder is not equal to 1.

3. Why is proving 3 as a quadratic non-residue of Mersenne primes significant?

Proving 3 as a quadratic non-residue of Mersenne primes has important implications in number theory and cryptography. It can be used to strengthen the security of certain cryptographic systems and also has connections to other unsolved problems in mathematics.

4. How is the proof of 3 as a quadratic non-residue of Mersenne primes achieved?

The proof involves using the properties of quadratic residues and non-residues, as well as the properties of Mersenne primes. It also requires advanced mathematical techniques such as number theory and algebraic number theory. The specific method used may vary depending on the specific Mersenne prime being considered.

5. Are there any other numbers that have been proven to be quadratic non-residues of Mersenne primes?

Yes, there are other numbers that have been proven to be quadratic non-residues of certain Mersenne primes. For example, it has been proven that 2 and 5 are quadratic non-residues of all Mersenne primes. However, the proof for 3 as a quadratic non-residue is significant because it holds for all Mersenne primes, regardless of their specific value.

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