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Integer m exists such that W(m)=W(m+1)=0

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December 6, 2010
algebrapolynomialmodular arithmeticquadraticsnumber theory proposednumber theory

Problem Statement

The polynomial W(x)=x2+ax+bW(x)=x^2+ax+b with integer coefficients has the following property: for every prime number pp there is an integer kk such that both W(k)W(k) and W(k+1)W(k+1) are divisible by pp. Show that there is an integer mm such that W(m)=W(m+1)=0W(m)=W(m+1)=0.
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