document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=557458>Question 557458</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mathematicians have been searching for a formula that yields prime numbers.  One such formula was x^2-x+41. Select some numbers for x, substitute them in the formula, and see if prime numbers occur.  Try to find a number for x that when substituted in the formula yields a composite number.  \r<BR>\n" );
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document.write( "I'm not sure of the steps to complete this problem...Please explain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #362674 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=362674\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=x%5E2-x%2B41 ALIGN=MIDDLE  ALT=\"P%28x%29=x%5E2-x%2B41\"   DISABLED_style= \"max-width:90%; height:auto;\" > was supposed to yield prime numbers.<BR>\n" );
document.write( "You are expected to try a few values for x , and find the corresponding P(x).<BR>\n" );
document.write( "It is likely to be a prime number. Here are a few:<BR>\n" );
document.write( "P(0)=41, P(1)=41, P(2)=43, P(3)=47, P(4)=53, P(5)=61, P(6)=71, P(7)=83, P(8)=97, P(9)=113, P(10)=131 P(20)=421, P(30)=971, P(40)=1601.<BR>\n" );
document.write( "All of those (and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29 ALIGN=MIDDLE  ALT=\"P%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > values in between) are prime numbers.<BR>\n" );
document.write( "It was a nice try, the design of P(x) ensured that iy could not be a multiple of 2, 3, 5, 7.<BR>\n" );
document.write( "However,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=x%5E2-x%2B41=x%28x-1%29%2B41 ALIGN=MIDDLE  ALT=\"P%28x%29=x%5E2-x%2B41=x%28x-1%29%2B41\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1 ALIGN=MIDDLE  ALT=\"x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > were a multiple of 41, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29 ALIGN=MIDDLE  ALT=\"P%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be a multiple of 41. So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2841%29=41%2A40%2B41=41%2A%2840%2B1%29=41%2A41 ALIGN=MIDDLE  ALT=\"P%2841%29=41%2A40%2B41=41%2A%2840%2B1%29=41%2A41\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2842%29=42%2A41%2B41=41%2A%2842%2B1%29=41%2A43 ALIGN=MIDDLE  ALT=\"P%2842%29=42%2A41%2B41=41%2A%2842%2B1%29=41%2A43\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2882%29=82%2A81%2B41=2%2A41%2A81%2B41=41%2A%282%2A81%2B1%29=41%2A163 ALIGN=MIDDLE  ALT=\"P%2882%29=82%2A81%2B41=2%2A41%2A81%2B41=41%2A%282%2A81%2B1%29=41%2A163\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2883%29=82%2A83%2B41=2%2A41%2A83%2B41=41%2A%282%2A83%2B1%29=41%2A167 ALIGN=MIDDLE  ALT=\"P%2883%29=82%2A83%2B41=2%2A41%2A83%2B41=41%2A%282%2A83%2B1%29=41%2A167\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and so on<BR>\n" );
document.write( "There are other values of P(x) that are not prime, too, like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2845%29=2021=43%2A47 ALIGN=MIDDLE  ALT=\"P%2845%29=2021=43%2A47\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2850%29=2491=81%2A53 ALIGN=MIDDLE  ALT=\"P%2850%29=2491=81%2A53\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2866%29=4331=61%2A71 ALIGN=MIDDLE  ALT=\"P%2866%29=4331=61%2A71\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2877%29=5893=71%2A83 ALIGN=MIDDLE  ALT=\"P%2877%29=5893=71%2A83\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2885%29=7181=43%2A167 ALIGN=MIDDLE  ALT=\"P%2885%29=7181=43%2A167\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );