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2015 China South East MO Grade 11 P8

Source: 2015 China South East MO Grade 11 P8

January 16, 2018
algebranumber theory

Problem Statement

Find all prime number pp such that there exists an integer-coefficient polynomial f(x)=xp−1+ap−2xp−2+…+a1x+a0f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0 that has p−1p-1 consecutive positive integer roots and p2∣f(i)f(−i)p^2\mid f(i)f(-i), where ii is the imaginary unit.
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