document.write( "Question 1203198: Hi. This is a base arithmetic question.
\n" ); document.write( "\"Is there any base b such that 3443 base b is a prime number? If yes, provide an example. If not, explain why not.\"
\n" ); document.write( "I've been stuck for a while. Any help on how to prove this would be appreciated. Thank you so much! \n" ); document.write( "

Algebra.Com's Answer #838562 by math_tutor2020(3817) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "3443 = 11 * 313\r
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\n" ); document.write( "\n" ); document.write( "This shows 3443 is not prime. It is composite.
\n" ); document.write( "It doesn't matter which base you are working with because we can convert between any two bases.
\n" ); document.write( "A prime number in one base, is a prime number in any base.\r
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\n" ); document.write( "\n" ); document.write( "Some examples:
\n" ); document.write( " $\"3443%5B5%5D+=+%2821%5B5%5D%29%2A%282223%5B5%5D%29\"$
\n" ); document.write( " $\"3443%5B7%5D+=+%2814%5B7%5D%29%2A%28625%5B7%5D%29\"$
\n" ); document.write( " $\"3443%5B4%5D+=+%2823%5B4%5D%29%2A%2810321%5B4%5D%29\"$
\n" ); document.write( "The subscript represents which base we're working in
\n" ); document.write( "For instance $\"3443%5B5%5D+=+matrix%281%2C3%2C3443%2C%22base%22%2C5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Further Reading:
\n" ); document.write( "https://math.stackexchange.com/questions/3999/is-a-prime-number-still-a-prime-when-in-a-different-base
\n" ); document.write( "and
\n" ); document.write( "http://web.archive.org/web/20190714164706/http://mathforum.org/library/drmath/view/55880.html\r
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\n" ); document.write( "\n" ); document.write( "Useful calculator
\n" ); document.write( "https://www.rapidtables.com/convert/number/base-converter.html\r
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\n" ); document.write( "\n" ); document.write( "Another approach\r
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\n" ); document.write( "\n" ); document.write( " $\"3443%5Bb%5D+=+3b%5E3+%2B+4b%5E2+%2B+4b+%2B+3\"$
\n" ); document.write( "Use the rational root theorem to determine that b = -1 is a root of $\"3b%5E3+%2B+4b%5E2+%2B+4b+%2B+3\"$
\n" ); document.write( "Therefore, (b+1) is a factor\r
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\n" ); document.write( "\n" ); document.write( "Use polynomial long division, or the shortcut synthetic division, to find that $\"%283b%5E3+%2B+4b%5E2+%2B+4b+%2B+3%29%2F%28b%2B1%29+=+3b%5E2%2Bb%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "That rearranges to $\"3b%5E3+%2B+4b%5E2+%2B+4b+%2B+3+=+%28b%2B1%29%283b%5E2%2Bb%2B3%29\"$
\n" ); document.write( "For $\"%28b%2B1%29%283b%5E2%2Bb%2B3%29\"$ to be prime, one of the factors must be 1.
\n" ); document.write( "If b+1 = 1, then b = 0. But we can't have base 0.\r
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\n" ); document.write( "\n" ); document.write( "If 3b^2+b+3 = 1, then it leads to two nonreal complex roots.
\n" ); document.write( "The base cannot be complex as only positive integers are allowed
\n" ); document.write( "Specifically from the set {5,6,7,8,9,...} so we can form $\"3443%5Bb%5D\"$ \r
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\n" ); document.write( "\n" ); document.write( "We conclude that neither factor (b+1) nor (3b^2+b+3) can be 1.
\n" ); document.write( "Therefore, $\"3b%5E3+%2B+4b%5E2+%2B+4b+%2B+3+=+%28b%2B1%29%283b%5E2%2Bb%2B3%29\"$ is never prime. It is always composite.
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