Problem 175

[ffff0]

Hi everyone.

While solving #175, I've found (x,96021946,x). I'm sure that this number is good, but it's not the answer. So there must be a smaller one.

Can anyone tell me how far from the truth I am? And, if possible, please tell me a number of 1s and 0s groups in the answer.

[hk]

I crossed out all correct digits.

[ffff0]

Thanks. At least I'm on a right track.

[rmacheshire]

While investigating problem 175 I calculated that f(242)=13 and f(241) = 17, using the function developed for problem 169. However the problem statement for 175 states that f(241) = 13 and f(240) =17. Which of us is right?

[DJohn]

While investigating problem 175 I calculated that f(242)=13 and f(241) = 17, using the function developed for problem 169. However the problem statement for 175 states that f(241) = 13 and f(240) =17. Which of us is right?

The problem statement is correct. It's possible that your implementation for problem 169 is wrong in general but happens to get the right result for that particular value.

[charre]

I use the decomposition in continued fractions of 13/17 which gives me directly Shortened Binary Expansion of n and by checking with the function of stern (or fusc of pb 169) I check my result which is correct; I applied to the requested fraction but the Shortened Binary Expansion found is not correct while I reconstitute the initial fraction well !!
is the input of the result correct with "," as a separator?
thanks for an answer
charre

[hk]

Yes, the problem as given is solved by 1859 users so far.

[charre]

Good morning
the statement of pb 175 asks for the smallest n such that f(n)/f(n-1)=a/b but we know mathematically that this n is unique because there is a bijection between the integers
positive and positive rational numbers using f(n)/f(n+1) provided that the fraction a/b is irreducible (which the proposed fraction is not) so I don't understand not the formulation of the statement "smallest n" !!
thanks for enlightening me

[jaap]

Good morning
the statement of pb 175 asks for the smallest n such that f(n)/f(n-1)=a/b but we know mathematically that this n is unique because there is a bijection between the integers
positive and positive rational numbers using f(n)/f(n+1) provided that the fraction a/b is irreducible (which the proposed fraction is not) so I don't understand not the formulation of the statement "smallest n" !!
thanks for enlightening me

This has nothing to do with bijections. Consider the example mentioned in the problem.

For instance, the smallest $n$ for which $f(n)/f(n-1)=13/17$ is $241$

$n$ must be a positive integer. For each of the values $n=1$, $n=2$, ..., $n=240$, if you calculate $f(n)/f(n-1)$ you get some rational number that is not equal to $13/17$, but if you calculate it for $n=241$ you find that $f(241)/f(240)=13/17$.
So $241$ is the smallest $n$ for which $f(n)/f(n-1)=13/17$. There is no smaller integer that works.