Problem 137

[Killa]

I found out that problem can be solved by finding first 15 integer terms of sequence sqrt(*snip*). I've tried to get then in many different ways, using c++ or mathematica, and always my algorithms are too slow. I could use some hint to find those, or some another idea to solve this problem.

Killa

[stijn263]

Can you perhaps spot a pattern among the first few values of n for which sqrt(*snip*) is integer?

Have fun solving!

[Killa]

To be honest i've seen this integers first time in my life (and im not new to math; )).

Friend helped me to solve it, so problem is already gone: ).

[hk]

If you're not new to math you've probably heard of Pell's equation.
In this case you're asked to find solutions to a Pell-like quadratic diophantine equation
So here are two keywords you can use to solve this kind of problem.
This kind of questions are not uncommon here.

[Killa]

Yeah, i've been thinking some about problem 140, kinda similiar to 137 and i found the connection with pell equasions, in some days i'll certainly find the answers and learn some new things.

Thanks for the hints: )

[GenePeer]

Can someone who solved the problem send me a PM explaining hyperdex's solution. In particular, I don't understand why he does this:

if the numerator is (~) mod (~), we check to see if it leads to a solution. If the numerator is (~) mod (~), we check to see if (~) the numerator leads to a solution. (Since we are looking for (~), we don't need to check any further multiples of the numerator.)

Why don't we need to check further multiples? His post is the twelfth on the first page. Thank you.

[Lord_Farin]

PM sent.

[RishadanPort]

I think there is some mistake on the wording of this question.

The question does say that we are considering only values of x such that Af(x) is a positive integer.

However, there are solutions for negative values of x.

IE x = -2 and x = 1/2 both result in Af(x) = 2


However for the sake of the problem, I guess that Af(x) = 2 is rational for both values of X, positive and negative

[RishadanPort]

I solved this problem...

An amazing problem, props to you guys

[DJohn]

IE x = -2 and x = 1/2 both result in Af(x) = 2

$A_F(-2) \neq 2$. It doesn't converge.

[RishadanPort]

Oh!

Sorry then. thanks for the reply.

[n2sz]

This is to confirm (I am pretty sure this is true) that the first nugget is the 1/2 explained in the problem statement. I have now found (bug in code) that was leading me to question whether this one wasn’t counted in the tally, but it is indeed nugget #1. Maybe that will help another lost soul…

Now off to find 11 thru 15…