1 Is The Largest Positive Integer

Algebra Level 1

Theorem. Among all positive integers, the integer 1 is the largest.

Step 1. Take any integer n 1 n \neq 1 .
Step 2. Since n 2 > n n^2 > n , it is not the largest positive integer.
Step 3. Therefore, 1 is the largest integer.

What is the error (if any) in the above proof?

Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. - Sherlock Holmes

We cannot take any integer n 1 n \neq 1 An arithmetic error when calculating n 2 > n n^2 > n It wrongly assumes the existence of a largest positive integer The conclusion is correct, there is no error in the proof

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1 solution

Calvin Lin Staff
Dec 5, 2016

The proof "works" by eliminating all possibilities for the largest positive integer other than n = 1 n = 1 , and then claiming that this remaining choice must hence be the largest positive integer.
However, what is has shown is just that "If there is a largest positive integer, then the only possibility for it (according to this proof) is n = 1 n = 1 ". As such, because there isn't a "largest positive integer", hence the result of "it is n = 1 n = 1 " does not follow.

Sherlock's quote applies to the context of his private investigation cases, where he knows that the scenario has existed and is thus looking for a reason to explain its existence. If so, by eliminating all but one possibility, we know that whatever remains must be the truth. It assumes existence through the real-world context that is implicit in his quote. A common scenario that happens is for people to assume that Brilliant problems are well-phrased, and thus there is a correct answer, so they can proceed by elimination.

In truth, this misconception of "If I'm left with 1 answer, then that is correct" occurs especially in geometry problems. The OP takes the algebraic interpretation, sets up the algebraic equation and solves for a certain value. However, it is possible for the answer to result in a physically impossible scenario. E.g. requires negative length which isn't accounted for, makes a certain angle reflex and so the wrong value is used, etc. This is why I often stress "Can this be achieved?" when reviewing a solution for mathematical rigor.

Here is an example where the solution assumes that there is a unique answer, and so merely calculates a necessary condition that gives a unique potential answer. For completeness, it has to show that this potential answer indeed satisfies the given initial conditions.

When I started to learn and discover the techniques of mathematics when I was a beginner,I imagined the same quote in my mind(for satisfactory purposes of a proof) that Sherlock did.Thanks for putting his quote here:)

Anandmay Patel - 4 years, 6 months ago

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In that case, please read the next 2 paragraphs, about when the logic is valid.

I'm a huge Sherlock Holmes fan :)

Calvin Lin Staff - 4 years, 6 months ago

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Some people like you write sentences which I wish goes on ad infinitum....

Anandmay Patel - 4 years, 6 months ago

But! The problem with this proof is not that it's illogical or presumptive rather it's incomplete. Since there's no testing of 1^2 to n^2. Since 1^2 < n^2 for all integers where 1<n the conclusion that 1 is largest is debunked.

Bob Smiley - 4 years, 6 months ago

The only issue I have with this problem is where you mention integer rather than natural number. The proof shows that, if there were a largest natural number it would have to be 1. However, among the integers, 0 contradicts the statement in Step 2.

Michael Childers - 4 years, 6 months ago

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In step 2,it is clearly written positive integer .

Anandmay Patel - 4 years, 6 months ago

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