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Number of Solutions to Two Polynomial Equations

Source:

January 12, 2009
algebrapolynomial

Problem Statement

For each positive integer m>1 m > 1, let P(m) P(m) denote the greatest prime factor of m m. For how many positive integers n n is it true that both P(n) \equal{} \sqrt{n} and P(n \plus{} 48) \equal{} \sqrt{n \plus{} 48}? <spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 4<spanclass=′latex−bold′>(E)</span> 5 <span class='latex-bold'>(A)</span>\ 0\qquad <span class='latex-bold'>(B)</span>\ 1\qquad <span class='latex-bold'>(C)</span>\ 3\qquad <span class='latex-bold'>(D)</span>\ 4\qquad <span class='latex-bold'>(E)</span>\ 5
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