Math0Phile

PROBLEM Let k be the smallest positive integer for which there exist distinct integers `m_1,\m_2,\m_3,\m_4,\m_5` such that the polynomial  p(x) = `(x-m_1)(x-\m_2)(x-\m_3)(x-\m_4)(x-\m_5)` has exactly k nonzero coefficients. Find, with proof, a set of integers `m_1,\m_2,\m_3,\m_4,\m_5` for which this minimum k is achieved. SOLUTION p(x) = `(x-m_1)(x-\m_2)(x-\m_3)(x-\m_4)(x-\m_5)`           = `x^5+Ax^4+Bx^2+Cx+D` where A, B, C, D are integers Now we want to minimise the number of nonzero coefficients that p(x) can have. So starting from the smallest positive integer, can k be 1 ? Well definitely no, because then p(x) = `x^5` `\Rightarrow` `m_1=m_2=m_3=m_4=m_5=0` Well then,  can k be 2 ? If k is 2, then p(x)  = `x^5+Cx` or p(x) = `x^5+D` as for any other case `x^2`| p(x)`\Rightarrow`any two of the `m_i` are zero and thus not distinct. If p(x)  = `x^5+Cx`, then p(x) = 0 will have one root as zero and the other four roots will come from ...