Parity Coefficient

If \( \dbinom{n}{r_n} \) is an even number for natural numbers \( r_n < n \) and denote \(R_n\) as the smallest possible value of \(r_n\), find the value of \(n\) less than \(10^4\) such that its corresponding \(R_n\) is maximized.

Details and Assumptions:

  • As an explicit example, \( {13 \choose 2}, {13 \choose 3}, {13 \choose 6}, {13 \choose 7}, {13 \choose 10}, {13 \choose 11} \) are all even numbers. So the possible values of \(r_{13} \) are \(2,3,6,7,10,11\). With \(R_{13} = 2 \).
  • Note that there doesn't exist some values of \(r_n \). For example, \( {7 \choose 1}, { 7 \choose 2} , \ldots , {7 \choose 6} \) are all odd numbers. So there's no value of \(r_7\) nor \(R_7 \).

Bonus: Can you generalize this for some constant \(M \) such that \(n < M \)?

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