What kinds of things can you do to prove that there is no largest prime number?

Answer 1

In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid in about 300 BC.
Since 1951 all the largest known primes have been found by computers. The search for ever larger primes has generated interest outside mathematical circles. The Great Internet Mersenne Prime Search and other distributed computing projects to find large primes have become popular in the last ten to fifteen years, while mathematicians continue to struggle with the theory of primes.

The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following:

Consider any finite set of primes. Multiply all of them together and add one (see Euclid number). The resulting number is not divisible by any of the primes in the finite set we considered, because dividing by any of these would give a remainder of one. Because all non-prime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes. Either way, there is at least one more prime that was not in the finite set we started with. This argument applies no matter what finite set we began with. So there are more primes than any given finite number.

This previous argument explains why the product P of finitely many primes plus 1 must be divisible by some prime not among those finitely many primes (possibly itself). This proof easily generalizes to any infinite unique factorization domain.

The proof is sometimes phrased in a way that leads the student to conclude that P + 1 must itself be prime, and think that Euclid's proof says the prime product plus 1 is always prime. The simple example of (2 · 3 · 5 · 7 · 11 · 13) + 1 = 30,031 = 59 · 509 (both primes) shows that this is not the case. In fact, any set of primes which does not include 2 will have an odd product. Adding 1 to this product will always produce an even number, which will be divisible by 2 (and therefore not be prime).

Other mathematicians have given other proofs. One of these (due to Euler) shows that the sum of the reciprocals of all prime numbers diverges. Another proof based on Fermat numbers was given by Goldbach.[3] Kummer's is particularly elegant and Harry Furstenberg provides one using general topology.[4][5]

Answer 2

Suppose you have a certain number N that you consider to be the largest prime number. Then, you take the product of all prime numbers up to N and add 1:

P = 1 * 2 * 3 * 5 * 7 * 11 * .... * N + 1

Now, whatever other number (prime or not) you divide P by, you will always have a residual of 1 - so therefore, P is another prime number.

Answer 3

Let's prove this without using any prime numbers. Let A be any non zero integer whatsoever. Let's make a sequence of numbers
which have no common factors among them. Then each
term will have primes which the other terms do not have.
So we start with A, A+1 which have no common factors.
Then we add the term A(A+1)+1 which has no common factors with A, or A+1. So now we have
A,A+1,A(A+1)+1, and we continue with A(A+1){A(A+1)+1}+1
and so on where each term is the product of the terms before
and then 1 is added to the result, except for the starter terms
A and A+1. Each term has it's very own primes and since there are infinitely many terms(potentially) then there are Infinitely many primes.

Answer 4

Stand at sea point and watch a tall deliver sail far off from you. It would not in basic terms get smaller and smaller, first the hull disappears, then the superstructure, then the decrease sails, then the top sails and ultimately the masthead, via fact it rather is disappearing over the curve of the Earth. in case you recognize the top of your eye, the top of the masthead and the fee of the vessel (which will desire to be consistent) you will have a stab at calculating the curvature, nevertheless i'm unlikely to attempt to describe the mathematics right here. Or degree the angular top of the sunlight at midday from you very own area and from 2 hundred miles north or south of your place. this will additionally help become attentive to the curvature. Or Watch the eclipse of the moon from diverse places. The Earth's shadow on the moon is often around. in basic terms a sphere can forged a around shadow in any direction.