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