document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204795>Question 1204795</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a,b,c,d, and e are distinct prime numbers such that: b=(a-1)^1/2 +1 ; c=(b-1)^1/2 +1 ; d=(c-1)^1/2 +1 ; e= (d-1)^1/2 +1 ; Find the least possible value of a+b+c+d+e<BR>\n" );
document.write( "A) 65537<BR>\n" );
document.write( "B) 65536<BR>\n" );
document.write( "C) 65816<BR>\n" );
document.write( "D) 65819<BR>\n" );
document.write( "E) 65814 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841292 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=841292\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'm assuming that b=(a-1)^1/2 +1 should be the equation b=(a-1)^(1/2) +1, which is the same as writing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=%28a-1%29%5E%281%2F2%29%5E%22%22+%2B1 ALIGN=MIDDLE  ALT=\"b=%28a-1%29%5E%281%2F2%29%5E%22%22+%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and converts to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+sqrt%28a-1%29%2B1 ALIGN=MIDDLE  ALT=\"b+=+sqrt%28a-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "List of the first few primes = {2,3,5,7,11,13,17,19,23,29,31,...}<BR>\n" );
document.write( "There are infinitely many prime numbers.\r<BR>\n" );
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document.write( "Let's say we picked the smallest prime for value \"e\".<BR>\n" );
document.write( "Doing so will guarantee that a+b+c+d+e is minimized.\r<BR>\n" );
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document.write( "e = (d-1)^(1/2) + 1<BR>\n" );
document.write( "(d-1)^(1/2) = e-1<BR>\n" );
document.write( "d-1 = (e-1)^2<BR>\n" );
document.write( "d = (e-1)^2 + 1<BR>\n" );
document.write( "d = (2-1)^2 + 1<BR>\n" );
document.write( "d = 2\r<BR>\n" );
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document.write( "e = 2 leads to d = 2<BR>\n" );
document.write( "But d must be different from e, so we have to go back to the drawing board.\r<BR>\n" );
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document.write( "Let's pick the next largest prime.<BR>\n" );
document.write( "e = 3<BR>\n" );
document.write( "d = (e-1)^2 + 1<BR>\n" );
document.write( "d = (3-1)^2 + 1<BR>\n" );
document.write( "d = 5<BR>\n" );
document.write( "We get a different value this time. So far so good.<BR>\n" );
document.write( "Furthermore, the result is prime.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "d = (c-1)^(1/2) + 1<BR>\n" );
document.write( "c = (d-1)^2 + 1<BR>\n" );
document.write( "c = (5-1)^2 + 1<BR>\n" );
document.write( "c = 17<BR>\n" );
document.write( "We get a different prime.\r<BR>\n" );
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document.write( "And,<BR>\n" );
document.write( "c = (b-1)^(1/2) + 1<BR>\n" );
document.write( "b = (c-1)^2 + 1<BR>\n" );
document.write( "b = (17-1)^2 + 1<BR>\n" );
document.write( "b = 257<BR>\n" );
document.write( "which is also prime. <BR>\n" );
document.write( "Primality can be checked using software. Or you can check the primes {2,3,5,7,11,13} to find they aren't factors of 257.\r<BR>\n" );
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document.write( "And lastly,<BR>\n" );
document.write( "b = (a-1)^(1/2) + 1<BR>\n" );
document.write( "a = (b-1)^2 + 1<BR>\n" );
document.write( "a = (257-1)^2 + 1<BR>\n" );
document.write( "a = 65537<BR>\n" );
document.write( "This value is prime.<BR>\n" );
document.write( "Use software to check primality.\r<BR>\n" );
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document.write( "To summarize<BR>\n" );
document.write( "a = 65537<BR>\n" );
document.write( "b = 257<BR>\n" );
document.write( "c = 17<BR>\n" );
document.write( "d = 5<BR>\n" );
document.write( "e = 3<BR>\n" );
document.write( "all of which are different primes and fit the equations mentioned.<BR>\n" );
document.write( "This is the smallest set possible since e = 3 was chosen among the smallest primes.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "a+b+c+d+e = 65537+257+17+5+3 = <font color=red size=4>65819</font> is the smallest sum possible, which leads to <font color=red size=4>answer choice D</font><BR>\n" );
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