Is this polynomial familiar? - III

Algebra Level 5

Let $k$ be a positive integer.

Consider the following condition:

There exists a polynomial $f(n)$ with rational coefficients such that for all positive integers $n<k$ , $f(n)$ is the $n-th$ digit after the decimal representation of $\pi$ .

For example, $f(1)=1, f(2)=4, f(3)=1, ...$

How many positive integers $k$ exist such that the above condition is false ?

The answer is 0.

1 solution

Abhinav Raichur
Dec 14, 2014

The question has been cleverly framed..... before starting to read further, ask yourselves a question... "can we frame arbitrary polynomials that yeild arbitrary answers with rational co-efficients?? ... the answer is yes

why? - consider we have a set of n values, we can fit it into a polynomial of degree n-1 { because of finite differences }

so the first digits of pi or any arbitrary digits of pi can be fitted into a polynomial.

*that could be interesting :) can we find some pattern in polynomials that frame pi? :) *

Is this polynomial familiar? - III

The answer is 0.

1 solution

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