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Turkey NMO 2006 1st Round - P26 (Number Theory)

Source:

February 3, 2013
modular arithmetic

Problem Statement

For how many primes pp, there exists an integr mm such that m3+3m−2≡0(modp)m^3+3m-2 \equiv 0 \pmod p and m2+4m+5≡0(modp)m^2+4m+5\equiv 0 \pmod p?
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 4<spanclass=′latex−bold′>(E)</span> Infinitely many <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}
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