Can You Find the Prime Factors of m Using Euler's Totient Function?

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In summary, if we know the numerical value of m and \Phi(m), we can use the formula S=m-Φ+1=p+q to deduce the prime factors p and q of m. This can be done by solving the quadratic equation x2 - Sx + m = 0, where x represents the prime factors. This method avoids the need for brute force calculations.
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squire636
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I understand that for m = pq where p and q are prime numbers, [itex]\Phi[/itex](m) = (p-1)(q-1). Is there any way that, knowing the numerical value of m and [itex]\Phi[/itex](m), we could deduce p and q, the prime factors of m?

Thanks!
 
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Specifically, I know that a huge number m is the product of two primes and I know [itex]\Phi[/itex](m)...but I can't figure out which primes those are and I don't want to figure it out by brute force.
 
  • #3
If you know m=pq and Φ=(p-1)(q-1), then define S=m-Φ+1=p+q.

So you have the product m and the sum S.
That means p and q are the solutions of the quadratic x2 - Sx + m = 0.
 

1. What is Euler's Totient Function?

Euler's Totient Function, also known as Euler's phi function, is a mathematical function used to determine the number of positive integers less than a given number that are relatively prime to that number.

2. How is Euler's Totient Function calculated?

Euler's Totient Function is calculated by taking the product of the given number and 1 minus the reciprocal of each prime factor of that number. For example, if the given number is 10, the prime factors are 2 and 5. Therefore, the calculation would be: 10 x (1 - 1/2) x (1 - 1/5) = 4.

3. What is the purpose of Euler's Totient Function?

Euler's Totient Function is used in number theory and cryptography to find the number of possible keys in a public key cryptography system. It is also used in other mathematical applications, such as determining the order of a group.

4. What are some key properties of Euler's Totient Function?

Some key properties of Euler's Totient Function include: it is multiplicative, meaning that for any two coprime numbers, the function value of their product is the product of their individual function values; it follows a recursive formula; and it is even for all numbers greater than 2.

5. How is Euler's Totient Function related to prime numbers?

Euler's Totient Function is closely related to prime numbers, as it takes into account the number of prime factors a given number has. If a number is prime, its totient function value is simply one less than the number itself. Additionally, the totient function value of two prime numbers multiplied together is equal to the product of their individual totient function values.

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