We interviewed a candidate and the senior quant cribbed at the response given by the candidate (although I found it rather adequate). Here is the problem and the solution. It was felt that the solution was not robust. I would like to know for myself the right solution. Question: N=1 is the largest positive natural number. Because if N > 1, then N^2 > N ( for N not equal to 1). Since there is a number N^2 greater than N (which is not 1), there is no other natural number except 1 that satisfies the rule that N^2 <= N. Hence, N =1 is the largest natural number.

"solution" to what? find the error in the "proof" that N=1 is the largest natural number, is that what the question is?

The error in the proof is the assumption that the largest natural number exists.

QuoteOriginally posted by: stilyo"solution" to what? find the error in the "proof" that N=1 is the largest natural number, is that what the question is?Yes Yes.I had the solution of the candidate typed earlier but removed it. Wel, the question is find flaw in the proof that N= 1 is the largest natural number.Because if N > 1 then N^2 > N and hence there is a number greater than N which contradicts that N >1 is the highest natural number. Therefore N =1 is the highest natural number. Find flaw in this proof.

QuoteOriginally posted by: ehThe error in the proof is the assumption that the largest natural number exists.quite true. now am trying to prove mathematically that this is indeed the error

well the argument can go like this:N is the largest positive natural number.Assume N is not equal to k.then (N-k)^2+k>k.There is no other natural number which satisfies (N-k)^2+k<=k except k.Therefore k is the largest natural number, and since k was arbitrary every natural number is the largest.Condition on a false premise (largest natural number exists), and you can prove anything. We just recently had a discussion along these lines about existence and uniqueness somewhere here.. what else is there to prove in a more formal manner I am not sure

QuoteOriginally posted by: stilyowell the argument can go like this:N is the largest positive natural number.Assume N is not equal to k.then (N-k)^2+k>k.There is no other natural number which satisfies (N-k)^2+k<=k except k.Therefore k is the largest natural number, and since k was arbitrary every natural number is the largest.Condition on a false premise (largest natural number exists), and you can prove anything. We just recently had a discussion along these lines about existence and uniqueness somewhere here.. what else is there to prove in a more formal manner I am not surePerpaps you nailed it although am somewhat not seeing the logical deduction of k being largest number satisfying (N-k)^2+k<=kThat being said, I remember vaguely from my real analysis courses about set of natural numbers N not being bounded (which is true). Hence, there is no least upper bound - no maximum.

QuoteOriginally posted by: iliketopologyPerpaps you nailed it although am somewhat not seeing the logical deduction of k being largest number satisfying (N-k)^2+k<=ki guess stylio meant k being the only natural number satisfying (N-k)^2+N <= N, so his example for k=1 would be N^2-N+1 instead of N^2 in your original post

QuoteOriginally posted by: wileyswQuoteOriginally posted by: iliketopologyPerpaps you nailed it although am somewhat not seeing the logical deduction of k being largest number satisfying (N-k)^2+k<=ki guess stylio meant k being the only natural number satisfying (N-k)^2+N <= N, so his example for k=1 would be N^2-N+1 instead of N^2 in your original postThat makes a lot of sense.

The error is using the argument to deduce that 1 is the largest natural number. What you've actually done is prove that the largest natural number cannot be anything other than 1. It's then trivial to prove that 1 is not the largest natual number either, and hence come to the conclusion that there is no largest natural number.

1) "Because if N > 1, then N^2 > N". Why?2) The assertion is "N natural number and maximal", so the contractiongives negation "not a Natural or not maximal"

QuoteOriginally posted by: iliketopologyWe interviewed a candidate and the senior quant cribbed at the response given by the candidate (although I found it rather adequate). Here is the problem and the solution. It was felt that the solution was not robust. I would like to know for myself the right solution. Question: N=1 is the largest positive natural number. Because if N > 1, then N^2 > N ( for N not equal to 1). Since there is a number N^2 greater than N (which is not 1), there is no other natural number except 1 that satisfies the rule that N^2 <= N. Hence, N =1 is the largest natural number.We dealing with natural numbers set. This set forms by define 0, 1 and operation called 'addition' not a 'power'. Thus for any N natural there exist N+1 which also is a natural number. On the other hand it is something lost in the introduced example. Indeed from the proof it follows that N = 1 is the unique natural number that satisfies N^2 <= N so what?

To be able to answer the question, I suspect that you need the axiomatic definition of the natural numbers, which is not trivial. Look up the Peano Axioms, and you will find the answer to this question.

QuoteOriginally posted by: listQuoteOriginally posted by: iliketopologyWe interviewed a candidate and the senior quant cribbed at the response given by the candidate (although I found it rather adequate). Here is the problem and the solution. It was felt that the solution was not robust. I would like to know for myself the right solution. Question: N=1 is the largest positive natural number. Because if N > 1, then N^2 > N ( for N not equal to 1). Since there is a number N^2 greater than N (which is not 1), there is no other natural number except 1 that satisfies the rule that N^2 <= N. Hence, N =1 is the largest natural number.We dealing with natural numbers set. This set forms by define 0, 1 and operation called 'addition' not a 'power'. Thus for any N natural there exist N+1 which also is a natural number. On the other hand it is something lost in the introduced example. Indeed from the proof it follows that N = 1 is the unique natural number that satisfies N^2 <= N so what?Like List said (and more specifically accordingly to the Peano Axioms), a property of the natural numbers is that if N is a natural number, then N+1 is also a natural number. With this property, you can prove there isnt a largest number, proving that the original claim in this thread to be wrong.