Finding large prime factors (number theory)

In summary, finding large prime factors involves identifying the prime numbers that divide a given integer without leaving a remainder. This process is crucial in number theory, particularly in cryptography, where the security of algorithms like RSA relies on the difficulty of factorizing large composite numbers into their prime components. Various algorithms, such as the Pollard rho algorithm, elliptic curve factorization, and the general number field sieve, have been developed to facilitate this task, each with varying degrees of efficiency based on the size of the numbers involved.
  • #1
bremenfallturm
57
11
Homework Statement
Find the prime factorizations of 314000.
Relevant Equations
Unsure - I've tried a couple of different methods as explained in the post
Hello!

I'm asked to factor 314000 by hand. The answer key says that is is ##2^4\cdot 5^3\cdot 157##, but I honestly have no idea how to factor it by hand.

I know that I can check all prime numbers up to ##\sqrt{314000}## but that would not be doable by hand obviously.
I did try to use the method that this video suggests:
(Which basically is: try to divide by 2, try to divide by 3, try to divide by 5, try to divide by 7)
But the problem is that ##157## is a factor.
How can I factor the number by hand?
 
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  • #2
Where is the problem?
$$
314000=314\cdot 1000=2\cdot 157 \cdot (10^3)=2\cdot 157\cdot (2\cdot 5)^3=2^4\cdot5^3\cdot 157
$$
 
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Likes berkeman, renormalize and gmax137
  • #3
Oh you're right. Sometimes you're just intimidated by a problem I guess. The problem is nowhere now.
 

FAQ: Finding large prime factors (number theory)

What is a prime factor?

A prime factor is a factor of a number that is a prime number, meaning it is greater than 1 and has no positive divisors other than 1 and itself. For example, the prime factors of 28 are 2 and 7.

Why is finding large prime factors important?

Finding large prime factors is crucial in various fields, particularly in cryptography. Many encryption algorithms, such as RSA, rely on the difficulty of factorizing large composite numbers into their prime components to secure data transmission.

What methods are used to find large prime factors?

Several methods can be employed to find large prime factors, including trial division, Pollard's rho algorithm, the quadratic sieve, and the general number field sieve. Each method has different efficiencies and is suitable for different sizes of numbers.

What is the significance of the RSA algorithm in relation to prime factors?

The RSA algorithm uses the product of two large prime numbers as a key for encryption and decryption. The security of RSA relies on the fact that while it is easy to multiply two large primes, it is computationally difficult to factor their product back into the original primes.

Are there any known algorithms that can efficiently find large prime factors?

While no polynomial-time algorithm is known for all cases, there are several efficient algorithms for specific cases, such as the elliptic curve factorization method and the general number field sieve, which is currently the fastest known method for factoring large integers.

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