Problem
NT-B2-M05-P016 The Central Coefficient Is Even
#16
★★★★★ Level 5 of 5
Prove that \(\binom{2n}{n}\) is even for every positive integer \(n\).
Let \(2^t\) be the largest power of two dividing \(n\). Look at the Legendre term with \(2^{t+1}\).
We need to prove \(v_2((2n)!)-2v_2(n!)>0\). Let \(n=2^t u\), where \(u\) is odd. In Legendre's sum, consider the term with \(2^{t+1}\): \(\lfloor2n/2^{t+1}\rfloor-2\lfloor n/2^{t+1}\rfloor=\lfloor u\rfloor-2\lfloor u/2\rfloor=1\). All other terms are nonnegative, so \(v_2\left(\binom{2n}{n}\right)\ge1\).
Strong problem: one must find a positive term, not compute everything.