Problem
NT-B2-M12-P008 Integer Values Without Integer Coefficients
#8
★★★☆☆ Level 3 of 5
Prove that \(P(n)=\frac{n(n-1)}2\) takes integer values for all integers \(n\), but \(P\notin\mathbb Z[x]\).
The product of two consecutive integers is even.
The numbers \(n\) and \(n-1\) have opposite parity, so their product is divisible by \(2\). Hence \(P(n)\in\mathbb Z\). However, the coefficient of \(x^2\) is \(\frac12\), not an integer. Thus \(P\) is not a polynomial with integer coefficients.
Useful for the later discussion of the binomial basis.