Are n1 and n2 Prime Factors of n?

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In summary, the conversation discusses a scenario where n is the smallest non-prime integer larger than or equal to 2, and it is not irreducible. This means that n can be written as n1.n2, where n1 and n2 are not units and are both greater than or equal to 2. The statement also mentions that n1 and n2 must have prime factors, but it is not clear how n > n1 and n > n2 implies this. It is concluded that if n is the smallest non-prime integer and n1 < n, this implies that n1 and n2 are prime factors.
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Homework Statement



We have n ≥ 2, n not prime, n ∈ ℤ. Take the smallest such n. n is not prime and as such n is not irreducible and can be written as n = n1.n2; n1, n2 not units. We may take n1, n2 ≥ 2. However we have n > n1, n > n2 so n1, n2 have prime factors.

I'm not sure how n > n1, n > n2 implies that n1, n2 have prime factors.

Homework Equations


I'm not sure what's relevant here.

The Attempt at a Solution



From what I can see, the lowest possible n which meets the criteria is 6. 6 has the prime factors 2 and 3, which means that obviously what is stated is true. I'm just not sure how n > n1, n > n2 implies that its true.
 
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6 is not the smallest non-prime integer larger than or equal to 2, 4 is.

Anyway, if n is the smallest non-prime integer and n1 < n, what does this imply?
 
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Do you mean, "n1, n2 are prime factors"?
 

FAQ: Are n1 and n2 Prime Factors of n?

What are prime numbers?

Prime numbers are positive integers that are only divisible by 1 and themselves. They have exactly two factors, making them unique among all positive integers.

How do we determine if a number is prime?

There are multiple methods for determining if a number is prime, including trial division, Sieve of Eratosthenes, and probabilistic tests such as Miller-Rabin. These methods involve checking if the number is divisible by other numbers or using mathematical properties of primes.

Are there an infinite number of prime numbers?

Yes, there are an infinite number of prime numbers. This was first proven by Euclid in 300 BC and is known as Euclid's theorem. The proof involves assuming a finite number of primes and then constructing a new prime number that is not included in that list.

What are some practical applications of prime numbers?

Prime numbers have many practical applications in fields such as cryptography, coding theory, and number theory. They are also used in the generation of random numbers and in the creation of secure communication protocols.

What is the largest known prime number?

As of 2021, the largest known prime number is 2^82,589,933 - 1, which has over 24 million digits. It was discovered in 2018 by the Great Internet Mersenne Prime Search (GIMPS) project.

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