But the doesn"t really occupational does it? What"s a very basic proof (no number concept or any kind of sort prefer that)?



Your proof is good. However, I check out some areas where you can add an ext detail:

Why space we assuming the $x$ is the largest herbal number? (Answer: because we room doing a proof by contradiction.)Why is the fact that $x + 1 > x$ relevant? (Answer: because it contradicts the fact that $x$ is the largest natural number.)

If we add these details in, the evidence looks like this:

We will display that there is no largest herbal number. In bespeak to perform this, let"s assume that $x$ is the largest natural number. Through the definition of "largest", over there is no herbal number i beg your pardon is higher than $x$. However, $x + 1$ is a natural number and $x + 1 > x$. This is a contradiction. QED.

You are watching: What is the largest natural number

There"s one an ext question that this proof doesn"t answer:

How carry out we know that $x + 1 > x$?

If friend haven"t currently proved the $x + 1 > x$, climate you must think around giving the a shot.

answer Dec 12 "16 in ~ 23:00

Tanner SwettTanner Swett
8,7692626 silver badges4949 bronze badges
add a comment |
I to be a viewpoint major, not dearteassociazione.orgematics, all the same, the complying with will prove what you space looking for:

Assume the visibility of a largest natural number (p)Being the largest natural number, "p" cannot be followed by a higher natural number. Meaning:

p + 1 ≤ p

Now subtract "p" from both sides. The result is:

1 ≤ 0

This has displayed that the presumption of "p" leader to a contradiction.

Therefore, that is not the situation that a largest natural number exists.


answer Nov 7 "17 at 14:10

3111 bronze argorial
include a comment |
Here is a constructive proof.

Let $P(n)$ it is in the residential property that over there exists a herbal number $n"$ wherein $n" > n$. We prove this by induction.

Base case - $n=1$: Clearly, $P(1)$, due to the fact that $484000 > 1$.

Step – assume for $n$, prove because that $n+1$: assume by induction theory that over there exists an $n""$ such the $n"" > n$. By the precongruence properties of $>$ we quickly see that $n""+484000 > n+1$. So take $n" = n"" + 484000$.

If we desire to prove the statement that it is not the instance that over there exists one $n_0 \in \dearteassociazione.orgbbN$ such the $n \leq n_0$ for every $n \in \dearteassociazione.orgbbN$, then us assume that there exists such an $n_0$. However then we have actually $n_0 + 484000 \leq n_0$, i m sorry is a contradiction. Therefore $n_0$ can not exist.

point out
edited Dec 12 "16 at 22:45
answered Dec 12 "16 at 22:42

Hans HüttelHans Hüttel
3,92022 gold badges1111 silver- badges2222 bronze badges
add a comment |
Maybe that can be displayed in a million the ways, but here is one proof just for fun (which surely lacks some foundational rigor). We will usage this facts:

1) $1$ is not the largest herbal number

2) because that every organic number $n \neq 1$ the number $n^2-n+1$ is a herbal number the is no equal to $n$ (that is, we have $n^2-n+1 \neq n$)

3) $(n-1)^2>0$ for every organic $n \neq 1$.

Suppose there exists largest natural number, contact it $n_L$, now kind a natural number $n_L^2-n_L+1$. Since $n_L$ is the largest and also because the 2) we have: $n_L>n_L^2-n_L+1$, which is equivalent to $0>(n_L-1)^2$ however this is not possible because $n_L \neq 1$ and because that 3).

point out
reply Dec 12 "16 at 23:09
4,86011 yellow badge1414 silver- badges3030 bronze title
include a comment |

your Answer

Thanks because that contributing solution to dearteassociazione.orgematics stack Exchange!

Please be certain to answer the question. Carry out details and also share her research!

But avoid

Asking for help, clarification, or responding to various other answers.Making statements based on opinion; earlier them up with referrals or personal experience.

Use dearteassociazione.orgJax to layout equations. Dearteassociazione.orgJax reference.

To learn more, watch our tips on writing an excellent answers.

See more: What Kinds Of Sealants Can I Put Polyurethane On Laminate Flooring Waterproof

Draft saved
Draft discarded

Sign up or log in

authorize up using Google
sign up making use of Facebook
sign up utilizing Email and also Password

Post together a guest

email Required, but never shown

Post together a guest


Required, but never shown

post Your prize Discard

By click “Post her Answer”, you agree come our regards to service, privacy policy and cookie plan

Not the prize you're looking for? Browse various other questions tagged number-theory discrete-dearteassociazione.orgematics or asking your very own question.

Featured ~ above Meta
Deep theorem v trivial proof
Is this a valid proof that there space infinitely countless natural numbers?
Does there exist a highly-composite number n > 2520 such that the next HCN is 2n?
permit n it is in an arbitrarily odd herbal number. Prove that $n^2≡1$ (mod4)
exactly how do i prove the "If prime ns does not divide organic number m, then gcd(p,m) = 1"
is there a huge list of extremely composite number?
What is the symbolic kind of "there does no exist a largest organic number "
Prove that the largest prime $
aid proving over there exists a natural number $k$ such that $a^k\equiv 1 (mod \space n)$
hot Network inquiries an ext hot questions

question feed
subscribe to RSS
question feed To i ordered it to this RSS feed, copy and paste this URL into your RSS reader.

ridge Exchange Network
site style / logo design © 2021 stack Exchange Inc; user contributions licensed under cc by-sa. Rev2021.10.15.40479

dearteassociazione.orgematics stack Exchange works finest with JavaScript allowed

her privacy

By clicking “Accept every cookies”, you agree ridge Exchange deserve to store cookies on your machine and disclose info in accordance v our Cookie Policy.