document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209349>Question 1209349</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of bases n \ge 2$ such that 100_n + 1_n is prime. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848584 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848584\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "100<sub>n</sub>+1<sub>n</sub> = 101<sub>n</sub> where n is an integer such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+%3E=+2+ ALIGN=MIDDLE  ALT=\"n+%3E=+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "101<sub>n</sub> = 101 base n\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To convert 101<sub>n</sub> to base 10 we will use this quadratic polynomial.<BR>\n" );
document.write( "1*n^2 + 0*n^1 + 1*n^0<BR>\n" );
document.write( "The coefficients 1,0,1 are from 101<sub>n</sub><BR>\n" );
document.write( "The exponents 2,1,0 count down by 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The expression<BR>\n" );
document.write( "1*n^2 + 0*n^1 + 1*n^0<BR>\n" );
document.write( "simplifies to <BR>\n" );
document.write( "n^2 + 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So 101<sub>n</sub> converts to n^2+1 base 10.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A few examples are<BR>\n" );
document.write( "101<sub>2</sub> = 2^2 + 1 = 5<sub>10</sub><BR>\n" );
document.write( "101<sub>3</sub> = 3^2 + 1 = 10<sub>10</sub><BR>\n" );
document.write( "101<sub>4</sub> = 4^2 + 1 = 17<sub>10</sub><BR>\n" );
document.write( "101<sub>5</sub> = 5^2 + 1 = 26<sub>10</sub><BR>\n" );
document.write( "101<sub>6</sub> = 6^2 + 1 = 37<sub>10</sub><BR>\n" );
document.write( "101<sub>7</sub> = 7^2 + 1 = 50<sub>10</sub><BR>\n" );
document.write( "101<sub>8</sub> = 8^2 + 1 = 65<sub>10</sub><BR>\n" );
document.write( "101<sub>9</sub> = 9^2 + 1 = 82<sub>10</sub><BR>\n" );
document.write( "101<sub>10</sub> = 10^2 + 1 = 101<sub>10</sub><BR>\n" );
document.write( "You can confirm each claim with a calculator such as <a  rel=nofollow HREF=\"https://www.rapidtables.com/convert/number/base-converter.html\">this one</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Of that list of examples we see the following primes in base ten: 5, 17, 37, 101<BR>\n" );
document.write( "Any prime in base 10 converts to a prime back in base n.<BR>\n" );
document.write( "For instance, 101<sub>4</sub> is prime (since 17<sub>10</sub> is prime). We cannot multiply smaller integer values in base 4 to arrive at 101<sub>4</sub>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the original problem is equivalent to asking: \"What integer values of n will make n^2+1 prime?\" \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As far as I know, the problem is unsolved in the mathematics community. Many topics about primes are also unsolved.<BR>\n" );
document.write( "Refer to <a  rel=nofollow HREF=\"https://math.stackexchange.com/questions/44126/primes-of-the-form-n21-hard\">this page</a> and <a  rel=nofollow HREF=\"https://oeis.org/A002496\">this page</a> for further discussion.<BR>\n" );
document.write( "The last link mentions \"It is conjectured that this sequence is infinite, but this has never been proved.\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It's quite possible that there might be a proof out there somewhere that I haven't found; or a proof could come along later in the future. I'll let another tutor chime in. <BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );