The first mathematician to do this may have been Euclid, but we can't be sure he was the first.
The following proof is due to Euler.
Let zeta(s) = sum of reciprocals of the s-th powers of the natural
numbers.
Because every number has a unique prime factorization, we have
zeta(s) = product over all primes p of (1+1/p^s+1/p^(2s) + ...)
which in turn can be written
zeta(s) = product over all prime p of 1/(1-1/p^s)
(an application of the formula for an infinite geometric series)
Note that the limit of zeta(1+e) as e tends to 0 is infinity.
This can be seen by making a comparison of the series of reciprocals
of the s-th powers of the naturals with an integral of 1/x^s.
If there were only finitely many primes, then this limit should
converge to the finite product over all p of 1/(1-1/p^s), a
contradiction.
Dirichlet generalized this idea to obtain his theorem that any
arithmetic series with base relatively prime to the common difference
contains infinitely many primes.
For another proof, you can show that the sums of the reciprocals of
the primes diverges...see Hardy and Littlewood's book on number
theory.
-Doctor Ceeks, The Math Forum.
Euclid
Yes, there is an infinite amount of prime numbers. This has been proven by the ancient Greek mathematician Euclid. As for composite numbers, since there are infinitely many natural numbers, there must also be an infinite amount of composite numbers, as they are all the natural numbers that are not prime.
Leonhard Euler
There are an infinite amount of prime numbers, as numbers never end. Prime numbers are numbers that are only divisible by 1 and itself. For example, 2, 3, 5, 53, and 97 are prime numbers.
Well, there is an infinite number of numbers, so technically, there is an infinite amount of prime numbers.
Euclid
Euclid
Yes, there is an infinite amount of prime numbers. This has been proven by the ancient Greek mathematician Euclid. As for composite numbers, since there are infinitely many natural numbers, there must also be an infinite amount of composite numbers, as they are all the natural numbers that are not prime.
Leonhard Euler
Euclid proved there are infinite. He said that if there were a finite number of primes, if you multiply all the primes together and then add 1, the result will be a prime. Thus, there are infinite primes.
They were known, at least, to the Ancient Greeks - perhaps earlier. I believe it was one of the Ancient Greeks who proved that the set of prime numbers was infinite (or "larger than any given set", or that there was no last prime number).They were known, at least, to the Ancient Greeks - perhaps earlier. I believe it was one of the Ancient Greeks who proved that the set of prime numbers was infinite (or "larger than any given set", or that there was no last prime number).They were known, at least, to the Ancient Greeks - perhaps earlier. I believe it was one of the Ancient Greeks who proved that the set of prime numbers was infinite (or "larger than any given set", or that there was no last prime number).They were known, at least, to the Ancient Greeks - perhaps earlier. I believe it was one of the Ancient Greeks who proved that the set of prime numbers was infinite (or "larger than any given set", or that there was no last prime number).
No, there are an infinite amount of non-prime numbers just as there are an infinite number of prime numbers.
Euclid was a Greek mathematician, and is called the father of geometry. A prime number is any number that can only be divided by itself and one. There are an infinite number of prime numbers. As a mathematician Euclid was interested the pursuit of knowledge for knowledge's sake and in proving that things could be quantified (how many, how much). In the case of primes the answer of "there are an infinitely large number of primes" was not available to him as the Greeks did not have the concept of "infinite".
There are infinite prime numbers as there is infinite numbers. You cannot limit the counting of primes.
There is an infinite amount of prime numbers all of which are odd numbers
There is an infinite set of prime numbers.
There is infinite amount of prime numbers. The largest known prime number is 243,112,609 − 1. It is a number with thirteen million digits. Greek mathematician Euclid proved it with the fact that if you multiply any given set of prime numbers and ad 1 you get either a prime number, or one that has smaller prime numbers - none of which is part of the original set. Example set: 2, 3, 5 2*3*5=30 30+1=31 Because of the added one, 31 is not divisible by 2, by 3, or by 5. (In this case, it happens to be a prime number.) For more info and more proofs that there is infinite number of prime numbers, check related link.