When do primes and integers in modulo 2^n form equal products?

  • Thread starter John Creighto
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In summary, the equation "P1^m1 mod 2^n = p2^m2 mod 2^n" is commonly used in scientific research for studying modular arithmetic. The value of n affects the complexity of the equation, while the values of p1, m1, p2, and m2 determine its result. In cryptography, the equation is used to generate secure keys, and it has various real-world applications in fields like computer science and engineering.
  • #1
John Creighto
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Let p1 and p2 be primes and m1 and m2 be integers when is:

When is p1^m1 mod 2^n = p2^m2 mod 2^n true?

I think this problem has applications to hash-tables.
 
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  • #2
They're equal when they're equal, there really isn't much to say. I don't see what being prime has to do with it.

As an abstract Abelian group, the odd numbers with multiplication are isomorphic to the product of the group with 2 elements and the cyclic group with 2n-2 elements.
 

FAQ: When do primes and integers in modulo 2^n form equal products?

What is the significance of "P1^m1 mod 2^n = p2^m2 mod 2^n" in scientific research?

The equation "P1^m1 mod 2^n = p2^m2 mod 2^n" is often used in scientific research to study the properties of modular arithmetic. It is a useful tool for analyzing patterns and relationships between numbers.

How does the value of n affect the equation "P1^m1 mod 2^n = p2^m2 mod 2^n"?

The value of n determines the size of the modulus, which is the number that the equation is being divided by. As n increases, the modulus becomes larger and the equation becomes more complex.

What is the relationship between the values of p1, m1, p2, and m2 in the equation "P1^m1 mod 2^n = p2^m2 mod 2^n"?

The values of p1, m1, p2, and m2 represent the base and exponent of each term in the equation. They are related to each other through the properties of modular arithmetic, and their values determine the result of the equation.

How is the equation "P1^m1 mod 2^n = p2^m2 mod 2^n" used in cryptography?

In cryptography, the equation is used to generate public and private keys for encryption and decryption. By using different values for p1, m1, p2, and m2, unique keys can be created for secure communication.

What are some real-world applications of the equation "P1^m1 mod 2^n = p2^m2 mod 2^n"?

The equation has numerous applications in fields such as computer science, physics, and engineering. It is used in error-correcting codes, data compression, and signal processing, among others. It is also used in the study of prime numbers and their properties.

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