Problem
NT-B2-M12-P016 A Polynomial of Given Degree with a Given Value
#16
★★★★☆ Level 4 of 5
Let \(N>1\), \(k>1\). Construct a polynomial \(p(x)\in\mathbb Z[x]\) of degree \(k\) and a positive integer \(m\) such that \(p(m)=N\).
1001 Problems in Classical Number Theory (method inspiration) · Problem 259
You may first choose \(m=1\), then adjust the constant term.
Take \(m=1\) and \(p(x)=x^k+N-1\). Then \(p\in\mathbb Z[x]\), its degree is \(k\), and \(p(1)=1+N-1=N\). The construction is complete.
A short construction problem; the point is not to overcomplicate it.