A Level number theory help

Original post by 0-)

I'm able to do the first two parts and I've shown that n = 1 mod 2 and n = 0 mod 5 but how can I use these results to find n? Or is it just a case of trial and error?

Naturally, given the famous 3^2 + 4^2 =5^2 relation, it is natural to check whether we have

33^2 + 44^2 = 55^2

And indeed we have. But you can narrow down your trial and error search quite a bit.

33^2 + 44^2 = n^2

means

33^2 = n^2 - 44^2

I.e.

33^2 = (n+44)(n-44)

LHS has repeated twice factors of 11 and 3, So the RHS must have the same.

44 already has 11 as a factor, so for the bracket (n+44) to be div by 11 we must have n being div by 11.

So already, n is odd, div by 5, and div by 11.

The smallest possibility which satisfies this is the number 55. The next one is 165 but clearly this cannot be the one since 165^2 is way too big.

We know that
n^2 = 33^2 + 44^2 < 50^2 + 50^2 = 5000
therefore n < sqrt(5000) < sqrt(6400) = 80 hence 165 is too large a number for it.

(edited 4 years ago)

Reply 2

Original post by RDKGames

Thank you!

So if such an n exists then it must be odd, div by 5 and div by 11 but has it been shown that an integer n must exist? You've limited it down to the only possibility which is 55 but how do you then know that n is actually 55 (as opposed to there being no solution)? Do you have to calculate 33^2 + 44^2 and check that it equals 55^2? If you do then that feels a bit silly because you could just find n like this without going through all this work.

Reply 3

Original post by 0-)

Indeed I have shown that our pool of numbers reduces to a single candidate which is 55.

To ensure that this is truly a solution, we need to show that (n+44)(n-44) for n=55 has the same prime decomposition as 33^2.

This bit is trivial.

Reply 4

David Getling

Original post by 0-)

I'm able to do the first two parts and I've shown that n = 1 mod 2 and n = 0 mod 5 but how can I use these results to find n? Or is it just a case of trial and error?

You've been told to do the question this way, so you must do so. But any real mathematician will be laughing their socks off at the suggestion that you do it this way. Here's the proper way do do questions like this.

For the first part use the fact that 33 ≡ 3 mod 5 or 33 ≡ -2 mod 5.

For the second part, number theory is the proverbial sledge hammer. You simply notice that 33² + 44² = 11²(3² + 4²)

Reply 5

Can someone please help me with this part?

I have shown that n^2 = 0 mod 2, n^2 = 0 mod 5 and n^2 = 1 mod 3, so by considering quadratic residues, n = 0 mod 2, n = 0 mod 5 and n = 1 mod 3 or n = 2 mod 3. Why does this prove that no integer solutions exist?

Reply 6

Can anyone help with the question above?

Reply 7

Original post by 0-)

Can someone please help me with this part?

Suppose such a solution exists.

Then n must be divisible by 2 and 5, but not 3. [due to your quad residues]

It must also be > 50 but < 70 (play around with the bounds). No such integer in this range. Hence no solution.

(edited 4 years ago)

Reply 8

Original post by RDKGames

Thanks again! I think my problem was that I expected the answer to be based on the number theory alone without having to check values. The process of checking values seemed a bit odd because you could just use a calculator in the first place to do the question.

Reply 9

Original post by 0-)

You've done the number theory bit anyway, and I'm not suggesting you use a calculator for this. I've given you a 3 liner answer using basic logic rather than computations.

It's about being able to reason logically why no such integer n exists.

Reply 10