1 and Prime Numbers - Numberphile

TL;DR

1 isn't a prime number due to unique prime factorization theorem.

Transcript

DR JAMES GRIME: We did a video. I'd done it some time ago about Mersenne primes. And offhandedly, just off the cuff, I said that 1 wasn't a prime. Now, if I'm honest, we don't say 1 is prime. There's a special reason for that. And we noticed that some people in the comments said, well, I think I've heard of this before, but why is that? Why isn't... Read More

Key Insights

• #️⃣ 1 was historically considered a prime number but was later excluded due to unique prime factorization requirements.
• #️⃣ The Fundamental Theorem of Arithmetic mandates every positive whole number to have a unique prime factorization.
• #️⃣ Prime numbers serve as the building blocks of all positive whole numbers through multiplication.
• #️⃣ Excluding 1 as a prime number eliminates ambiguity in prime factorization and simplifies mathematical concepts.
• 🏪 The exclusion of 1 as a prime number is a choice made for mathematical convenience and clarity.
• #️⃣ Prime numbers play a crucial role in mathematics, similar to atoms in chemistry, essential for understanding number theory.
• ❓ The concept of unique prime factorization ensures consistency and clarity in mathematical calculations.

Questions & Answers

Q: Why isn't 1 considered a prime number?

1 isn't prime because the Fundamental Theorem of Arithmetic requires unique prime factorization, which 1 doesn't fit with multiple ways of being expressed as a product.

Q: How does the exclusion of 1 as a prime affect mathematics?

Excluding 1 as a prime ensures a clear and unique way to express numbers as products of primes, simplifying mathematical theorems and calculations.

Q: Why does the exclusion of 1 as a prime help in defining prime numbers?

By excluding 1 as a prime, mathematicians ensure that prime numbers serve as building blocks for other numbers, maintaining the fundamental principle of prime factorization.

Q: Can 1 be considered a prime in certain contexts?

While mathematicians have chosen to exclude 1 as a prime in most instances, the definition of primes is flexible, allowing for inclusion in specific scenarios if deemed necessary.

Summary & Key Takeaways

• Historically, 1 was considered a prime number but was later excluded due to the Fundamental Theorem of Arithmetic.

• The theorem states that every positive whole number can be uniquely written as a product of prime numbers.

• Excluding 1 as a prime avoids ambiguity in prime factorization.