document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=476630>Question 476630</A>: <I><SPAN STYLE=\"word-break: break-all;\"> \"if (2^k)-1 is a prime number then, prove that (2^(k-1))(2^k-1)is a perfect number.\"<BR>\n" );
document.write( "(i know it is the definition but i want proof) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #326822 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=326822\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Theorem:<BR>\n" );
document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2k-1 ALIGN=MIDDLE  ALT=\"2k-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=prime ALIGN=MIDDLE  ALT=\"prime\"   DISABLED_style= \"max-width:90%; height:auto;\" > number, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2k-1%282k-1%29 ALIGN=MIDDLE  ALT=\"2k-1%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perfect ALIGN=MIDDLE  ALT=\"perfect\"   DISABLED_style= \"max-width:90%; height:auto;\" > number and every even perfect number has this form.\r<BR>\n" );
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document.write( "Proof:  \r<BR>\n" );
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document.write( "Suppose first that  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+2k-1 ALIGN=MIDDLE  ALT=\"p+=+2k-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=prime ALIGN=MIDDLE  ALT=\"prime\"   DISABLED_style= \"max-width:90%; height:auto;\" > number, and set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+2k-1%282k-1%29 ALIGN=MIDDLE  ALT=\"n+=+2k-1%282k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "To show <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perfect ALIGN=MIDDLE  ALT=\"perfect\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need only show <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma%28n%29+=+2n ALIGN=MIDDLE  ALT=\"sigma%28n%29+=+2n\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" > is multiplicative and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma%28p%29+=+p%2B1+=+2k ALIGN=MIDDLE  ALT=\"sigma%28p%29+=+p%2B1+=+2k\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we know\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma%28n%29+=+sigma%282k-1%29.sigma%28p%29+=++%282k-1%292k+=+2n ALIGN=MIDDLE  ALT=\"sigma%28n%29+=+sigma%282k-1%29.sigma%28p%29+=++%282k-1%292k+=+2n\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "This shows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perfect ALIGN=MIDDLE  ALT=\"perfect\"   DISABLED_style= \"max-width:90%; height:auto;\" > number.\r<BR>\n" );
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document.write( "    On the other hand, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=suppose ALIGN=MIDDLE  ALT=\"suppose\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n ALIGN=MIDDLE  ALT=\"+n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is any <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=even ALIGN=MIDDLE  ALT=\"even\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+perfect ALIGN=MIDDLE  ALT=\"+perfect\"   DISABLED_style= \"max-width:90%; height:auto;\" > number and write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2k-1m ALIGN=MIDDLE  ALT=\"2k-1m\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=odd ALIGN=MIDDLE  ALT=\"odd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+integer ALIGN=MIDDLE  ALT=\"+integer\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E2 ALIGN=MIDDLE  ALT=\"k%3E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "Again sigma is multiplicative so\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma%282k-1m%29+=+sigma%282k-1%29.sigma%28m%29+=+%282k-1%29.sigma%28m%29 ALIGN=MIDDLE  ALT=\"sigma%282k-1m%29+=+sigma%282k-1%29.sigma%28m%29+=+%282k-1%29.sigma%28m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perfect ALIGN=MIDDLE  ALT=\"perfect\"   DISABLED_style= \"max-width:90%; height:auto;\" > we also know that\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma%28n%29+=+2n+=+2km ALIGN=MIDDLE  ALT=\"sigma%28n%29+=+2n+=+2km\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Together these two criteria give\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2km+=+%282k-1%29.sigma%28m%29 ALIGN=MIDDLE  ALT=\"2km+=+%282k-1%29.sigma%28m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
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document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2k-1 ALIGN=MIDDLE  ALT=\"2k-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > divides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2km ALIGN=MIDDLE  ALT=\"2km\"   DISABLED_style= \"max-width:90%; height:auto;\" > hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2k-1 ALIGN=MIDDLE  ALT=\"2k-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > divides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >, say \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%282k-1%29M ALIGN=MIDDLE  ALT=\"m+=+%282k-1%29M\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "Now substitute this back into the equation above and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=divide ALIGN=MIDDLE  ALT=\"divide\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2k-1 ALIGN=MIDDLE  ALT=\"2k-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2kM+=+sigma%28m%29 ALIGN=MIDDLE  ALT=\"2kM+=+sigma%28m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" > are both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=divisors ALIGN=MIDDLE  ALT=\"divisors\"   DISABLED_style= \"max-width:90%; height:auto;\" > of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > we know that\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2kM+=+sigma%28m%29+%3E+m+%2B+M+=+2kM ALIGN=MIDDLE  ALT=\"2kM+=+sigma%28m%29+%3E+m+%2B+M+=+2kM\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
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document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma%28m%29+=+m+%2B+M ALIGN=MIDDLE  ALT=\"sigma%28m%29+=+m+%2B+M\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=prime ALIGN=MIDDLE  ALT=\"prime\"   DISABLED_style= \"max-width:90%; height:auto;\" > and its <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=only ALIGN=MIDDLE  ALT=\"only\"   DISABLED_style= \"max-width:90%; height:auto;\" > two <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=divisors ALIGN=MIDDLE  ALT=\"divisors\"   DISABLED_style= \"max-width:90%; height:auto;\" > are itself (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and one (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" >).  \r<BR>\n" );
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document.write( "Thus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+2k-1 ALIGN=MIDDLE  ALT=\"m+=+2k-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=prime ALIGN=MIDDLE  ALT=\"prime\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=prove ALIGN=MIDDLE  ALT=\"prove\"   DISABLED_style= \"max-width:90%; height:auto;\" > that the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > has the prescribed form. \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );