document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209684>Question 1209684</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all integers n for which \frac{n^2 + n + 1}{n - 2 + n^3} is an integer.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849805 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849805\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the given expression be equal to an integer $k$.<BR>\n" );
document.write( "\[ \frac{n^2 + n + 1}{n^3 + n - 2} = k \]<BR>\n" );
document.write( "\[ n^2 + n + 1 = k(n^3 + n - 2) \]<BR>\n" );
document.write( "\[ n^2 + n + 1 = kn^3 + kn - 2k \]<BR>\n" );
document.write( "\[ kn^3 - n^2 + (k-1)n - (1+2k) = 0 \]\r<BR>\n" );
document.write( "\n" );
document.write( "If $k=0$, then $n^2 + n + 1 = 0$, which has no integer solutions.\r<BR>\n" );
document.write( "\n" );
document.write( "If $n=0$, then $\frac{1}{-2} = k$, which is not an integer.<BR>\n" );
document.write( "If $n=1$, then $\frac{3}{0}$, which is undefined.<BR>\n" );
document.write( "If $n=2$, then $\frac{7}{8}$, which is not an integer.<BR>\n" );
document.write( "If $n=3$, then $\frac{13}{28}$, which is not an integer.\r<BR>\n" );
document.write( "\n" );
document.write( "If $n$ is large, then $n^3$ dominates, so $k$ must be close to 0.\r<BR>\n" );
document.write( "\n" );
document.write( "If $k=1$, we have $n^3 - n^2 + 0n - 3 = 0$.<BR>\n" );
document.write( "$n^3 - n^2 - 3 = 0$.<BR>\n" );
document.write( "If $n=2$, $8-4-3 = 1 \ne 0$.<BR>\n" );
document.write( "If $n=3$, $27-9-3 = 15 \ne 0$.\r<BR>\n" );
document.write( "\n" );
document.write( "If $n^3 + n - 2 = 1$, then $n^3 + n - 3 = 0$.<BR>\n" );
document.write( "If $n=1$, $1+1-3 = -1$.<BR>\n" );
document.write( "If $n=2$, $8+2-3 = 7$.\r<BR>\n" );
document.write( "\n" );
document.write( "If $n^3 + n - 2 = -1$, then $n^3 + n - 1 = 0$.<BR>\n" );
document.write( "If $n=0$, $-1$.<BR>\n" );
document.write( "If $n=1$, $1$.\r<BR>\n" );
document.write( "\n" );
document.write( "If $n=2$, $n^3+n-2 = 8+2-2 = 8$. Then $\frac{4+2+1}{8} = \frac{7}{8}$.\r<BR>\n" );
document.write( "\n" );
document.write( "Consider the case where $n$ is large. Then $k \approx \frac{n^2}{n^3} = \frac{1}{n}$.<BR>\n" );
document.write( "If $n$ is large, $k$ must be close to 0.\r<BR>\n" );
document.write( "\n" );
document.write( "If $k=0$, $n^2+n+1=0$, no integer solution.\r<BR>\n" );
document.write( "\n" );
document.write( "If $n=1$, $\frac{3}{0}$ undefined.<BR>\n" );
document.write( "If $n=2$, $\frac{7}{8}$.<BR>\n" );
document.write( "If $n=3$, $\frac{13}{28}$.\r<BR>\n" );
document.write( "\n" );
document.write( "If $n=-1$, $\frac{1-1+1}{-1-1-2} = \frac{1}{-4}$.<BR>\n" );
document.write( "If $n=-2$, $\frac{4-2+1}{-8-2-2} = \frac{3}{-12} = -\frac{1}{4}$.<BR>\n" );
document.write( "If $n=-3$, $\frac{9-3+1}{-27-3-2} = \frac{7}{-32}$.\r<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: The final answer is $\boxed{no solutions}$<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );