Understanding Euclid's proof that the number of primes is infinite ...
2016年4月7日 · $\begingroup$ Here we have the usual historical mistake, dating back at least to Dirichelt, which holds that Euclid's proof was by contradiction. Euclid's actual proof was simpler and better than that. See my answer below. $\qquad$ $\endgroup$ –
elementary number theory - About Euclid's proof of infinite primes ...
$\begingroup$ No, Euclid's proof is constructive (not by contradiction) and it does give an algorithm to generate an infinite list of primes. It shows how given any finite set of prime one can construct a new prime not in that set. $\endgroup$
Is there an intuitionist (i.e., constructive) proof of the infinitude ...
2011年7月11日 · $\begingroup$ I would describe Euclid's proof as indirect and constructive. The logic behind the construction of a prime depends on the premise that you have already listed the complete finite set of primes. For example, applying Euclid's algorithm to {2,3,5,7,11,13}, you get the 'prime' 30,031 = 59 * 509. $\endgroup$ –
Euclid's proof of the infinitude of primes to prove this question
Just like in Euclid's proof, you can't possibly prove that $1+\prod p_{i}$ is prime. Same here, you can't possibly prove that what you get is irreducible, because it might as well isn't. For example, $1+x(x+1)$ is not irreducible in $\mathbb{C}$.
Is Euclid's proof on the infinitude of primes flawed because it …
$\begingroup$ @alexis: What I meant by "Euclid's proof fails..." is only that (1) saying that Euclid proves the set of all primes to infinite is a misrepresentation, because notion of "set", "all primes" and "infinite" are anachronisms with respect to Euclid (2) if you must interpret what Euclid does write into set theory, then it is closest to "there are more primes than in any given set of ...
Proof of infinitely many prime numbers - Mathematics Stack …
2016年10月9日 · So the proof works. In fact, it works precisely the same regardless of what set of numbers we suppose are the only primes that exist. Thus no finite set of numbers can include all the primes that exist. Thus there are an infinite number of primes.
elementary number theory - Euclid's proof of infinitude of primes ...
This has infinite primes but not unique factorization. I cannot think of an example that has an infinite number of elements and a total ordering, does have unique factorization, yet does not have an infinite set of primes. But if this can't exist, the proof is probably at least as complex as the Euclid proof that primes are infinite in $\Bbb{N}$.
Infinitude of Primes proof. - Mathematics Stack Exchange
2023年10月7日 · $\begingroup$ I'm not really sure what you think is "new" about this approach. This is exactly how the usual proof (Euclid's) goes: given any finite set of primes, it explicitly constructs a "gap number" for it (or in fact something stronger, a number that is not even divisible by any of the primes in your set).
elementary number theory - Modified Euclid's proof of infinite …
Q. Alternate the proof for Euclid's infinite number of primes to show there are infinitely prime numbers of the form $6n-1$ where n is an integer. my attempt, suppose by contradiction there are
number theory - Euclid's proof on the infinity of primes
2015年9月12日 · I perfectly understand Euclid's proof on the infinity of primes. Let's suppose there is a largest prime, p, and then let's make a number, n, so that n = (2 x 3 x 5 x ... x p) + 1. ( The parentheses are not mandatory but they help me to visualize it better.)