Table of Contents
What is the hardest prime number?
208003
2. The “Top Ten”
rank | prime | when |
---|---|---|
1 | 208003!-1 | 2016 |
2 | 150209!+1 | 2011 |
3 | 147855!-1 | 2013 |
4 | 110059!+1 | 2011 |
Is there an infinity of primes?
The Infinity of Primes. The number of primes is infinite. The first ones are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 and so on. The first proof of this important theorem was provided by the ancient Greek mathematician Euclid.
What is the best proof that there are infinitely many primes?
Euclid’s Proof of the Infinitude of Primes (c. 300 BC) Euclid may have been the first to give a proof that there are infinitely many primes. Even after 2000 years it stands as an excellent model of reasoning.
How many primes are there?
There are infinitely many primes. Proof. Suppose that p 1=2 < p 2 = 3 < < p r are all of the primes. Let P = p 1p 2…p r+1 and let p be a prime dividing P; then p can not be any of p 1, p 2., p r, otherwise p would divide the difference P-p 1p 2…p r=1, which is impossible.
How do you find the divisibility of a list of primes?
Call the primes in our finite list p1, p2., pr . Let P be any common multiple of these primes plus one (for example, P = p1p2 pr +1). Now P is either prime or it is not. If it is prime, then P is a prime that was not in our list. If P is not prime, then it is divisible by some prime, call it p.
How many prime numbers are there in topology?
There are infinitely many primes. Proof. Define a topology on the set of integers by using the arithmetic progressions (from -infinity to +infinity) as a basis. It is easy to verify that this yields a topological space. For each prime p let Ap consists of all multiples of p .