document.write( "Question 1208953: Suppose a >= 2 and n is a natural number larger than 1.
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Algebra.Com's Answer #847498 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "Suppose a is a natural number >= 2 and n is a natural number larger than 1.
\n" ); document.write( "How can I prove that if n is odd, then $\"a%5En%2B1\"$ is not a prime?
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\n" ); document.write( "\n" ); document.write( " I edited your post - see the underlined fragment - to make it precisely correct.\r
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document.write( "If n is an odd positive integer number greater than 1, then there is a decomposition\r\n" );
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document.write( "     = \r\n" );
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document.write( "(the signs at degrees of \"a\" alternate \" + \" and \" - \").\r\n" );
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document.write( "It is a well known formula. To prove it, it is enough to open parentheses.\r\n" );
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document.write( "This formula shows and tells that every integer number of the form   \r\n" );
document.write( "with integer positive  \" a \"  and natural odd  n > 1  is a composite number.\r\n" );
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document.write( "It is what you want to prove.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "For n=3, the decomposition $\"a%5E3%2B1\"$ = $\"%28a%2B1%29%2A%28a%5E2+-+a+%2B+1%29\"$ is studied explicitly
\n" ); document.write( "in standard school Math curriculum as the sum of cubes decomposition.\r
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\n" ); document.write( "\n" ); document.write( "For other odd natural integer \" n \" greater than 3, it can be
\n" ); document.write( "easily obtained from the formula of the sum of a geometric progression\r
\n" ); document.write( "\n" ); document.write( " 1, -a, a^2, -a^3, a^4, -a^5, . . . , a^(n-1)\r
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\n" ); document.write( "\n" ); document.write( "with alternate signs, which corresponds to the case of common ratio \" -a \".\r
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