Determine the smallest value

Troubles again (Giggle)

Determine the smallest value of ; m and n are positive integers.

I created a graph of it and as far as I've noticed 11 is the smallest value.

The thing is- how to prove it?

I tried to play with divisibility by 11, but I didn't succeed.

Any ideas?

Regards,

Lukasz

Re: Determine the smallest value

Quote:

Originally Posted by

**Lukaszm** Troubles again (Giggle)

Determine the smallest value of

; m and n are positive integers.

I created a graph of it and as far as I've noticed 11 is the smallest value.

The thing is- how to prove it?

I tried to play with divisibility by 11, but I didn't succeed.

Any ideas?

Regards,

Lukasz

I belive that your answer is correct.

we can write twenty as

If we expand using the binomial theorem we get

Now if we subtract we get

If we set m=n the first two terms reduce out.

Now since all terms of the sum have the same sign we must minimize the sum. the value m=1 gives this minimum

This should give you the conclustion that you are looking for.

Re: Determine the smallest value

Here's a proof that k = 11.

Let s.t. is a min among such numbers, .

Look at it modulo 10: .

Thus .

Since , and is defined to be minimal, conclude that .

But 9 can be excluded, for if , then , which is impossible (look , recall ).

Thus .

Also, you can exclude , because:

Assume . Then , which implies .

But 19 divides , so 19 divides , a contradiction. Thus .

Thus .

Need to show that has no solutions, and then 11 will be the answer.

Look mod 10 again, and see that n must be even. Thus need only show that has no solutions, and then 11 will be the answer.

Look at using mod 19: , so

Now , have: .

Filling in:

and .

Therefore has no solutions.

Therefore has no solutions.

Therefore has no solutions.

Therefore, k = 11.