document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=500690>Question 500690</A>: <I><SPAN STYLE=\"word-break: break-all;\"> One of the most famous unsolved problems in mathematics is a conjecture<BR>\n" );
document.write( "made by Christian Goldbach in a letter to Leonhard Euler in 1742. The<BR>\n" );
document.write( "conjecture made in this letter is now known as Goldbach’s Conjecture. The<BR>\n" );
document.write( "conjecture is as follows:\r<BR>\n" );
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document.write( "Every even integer greater than 2 can be expressed as the sum of two<BR>\n" );
document.write( "(not necessarily distinct) prime numbers.\r<BR>\n" );
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document.write( "Currently, it is not known if this conjecture is true or false, although most<BR>\n" );
document.write( "mathematicians believe it to be true.<BR>\n" );
document.write( "(a) Describe one way to prove that Goldbach’s Conjecture is false.<BR>\n" );
document.write( "(b) Prove the following:<BR>\n" );
document.write( "If there exists an odd integer greater than 5 that is not the sum of<BR>\n" );
document.write( "three prime numbers, then Goldbach’s Conjecture is false. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #338234 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=338234\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "1.  Find an even integer greater than 2 that cannot be expressed as the sum of two prime numbers.\r<BR>\n" );
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document.write( "2.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\"> be an odd integer greater than 5 such that three not necessarily distinct prime numbers that sum to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\"> do not exist.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> be a prime number greater than 2, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\"> must be odd.  The difference of any two odd integers is even <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left((2m\ -\ 1\)\ -\ \left(2n\ -\ 1\right)\ =\ (2m\ -\ 2n)\right)\">.  Hence, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\ -\ p\"> is even.  But Goldberg's conjecture says that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\ -\ p\"> must be the sum of two primes, and therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\"> must be the sum of three primes.  Therefore, if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\"> exists, there must be a <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\ -\ p\"> that is even and not the sum of two primes proving Goldberg's Conjecture false.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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