Q: Are there an infinite number of prime numbers?

Physicist: Yes.  Here’s a proof (there are many):
1) Assume there are a finite number of primes.
2) Multiply them all together and add 1.
3) This new number is not divisible by any of the original primes so it must be a new prime (or be divisible by at least one new prime).

This means that no matter how many primes exist, there must be at least one more.  But that’s a one way trip to infinity.
This is a contradiction, so the assumption that there are a finite number of primes is hereby debunked.
There are actually dozens of different proofs of “the infinitude of primes” but this one is probably the simplest.

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8 Responses to Q: Are there an infinite number of prime numbers?

  1. David says:

    My favorite is Euclid’s proof. It’s a bit longer, but it might satisfy some skeptics out there…

  2. Pingback: Q: How do you talk about the size of infinity? How can one infinity be bigger than another? | Ask a Mathematician / Ask a Physicist

  3. tanuj says:

    dear physicist all the prime nos. are odd and odd*odd = odd and if u add 1 to it it will make it even and non prime respectively ,there is an authentic euclid concept we don`t want yours.

  4. Dylan says:

    tanuj, But when you add one, you’ll notice that one of the factors is two, a prime number you forgot to consider.

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  6. Clyde says:

    I accept there are proofs for infinite primes. My question is why do we search for them if they are infinite?

    The effort required to find what are now galactic-sized primes (>17 million digits) requires some serious computing resources. Why bother? And why does it make the news each time a new one is found every 2-5 years? ~yawn~ Tell me again why I’m supposed to be excited?

    Actually, I do think it’s exciting and primes are cool. I just treat the infinity aspect for what it is — buzz kill.

  7. Hosting says:

    In 1874, in his first set theory article , Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable.

  8. Pingback: Q: How can something be “proven” in science or math? | Ask a Mathematician / Ask a Physicist

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