document.write( "Question 966690: For a prime p>3, prove that p^2 - 1 is divisible by 12\r
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Algebra.Com's Answer #592047 by Edwin McCravy(20056) $\"\"$ $\"About$

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For a prime p>3, prove that p^2 - 1 is divisible by 12
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document.write( "For any prime p>3, p^2-1 > 6.\r\n" );
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document.write( "All odd numbers > 6 are between 2 consecutive multiples of 6.\r\n" );
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document.write( "Between every 2 consecutive multiples of 6, there are 3 \r\n" );
document.write( "consecutive odd numbers, 6k+1, 6k+3, and 6k+5. I.e.,\r\n" );
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document.write( "6k < 6k+1 < 6k+3 < 6k+5 < 6k+6=6(k+1)\r\n" );
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document.write( "Thus every odd number > 6 can be written as 6k+1, 6k+3 or 6k+5.\r\n" );
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document.write( "As all primes greater than 3 are odd, they can all be written \r\n" );
document.write( "as 6k+1 or 6k+5, (but never as 6k+3 since that is never prime).\r\n" );
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document.write( "(6k+1)²-1 = 36k²+12k+1-1 = 36k²+12k    <-- a multiple of 12\r\n" );
document.write( "(6k+5)²-1 = 36k²+60k+25-1 = 36k²+60k+24 <-- a multiple of 12\r\n" );
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document.write( "That proves it.\r\n" );
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document.write( "Edwin

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