document.write( "Question 849360: Prove that every prime of the form 3m + 1 with m (in) N is also of the form 6n + 1
\n" ); document.write( "with n (in) N.
\n" ); document.write( "N for natural numbers \n" ); document.write( "
Algebra.Com's Answer #511505 by LinnW(1048)\"\" \"About 
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We need to show that if p is a prime number of the form
\n" ); document.write( "6n + 1 , then there exists an m such that 3m + 1 = p.
\n" ); document.write( "We also need to show that if p is a prime number of the form
\n" ); document.write( "3m + 1, then there exists an n such that 6n + 1 = p.
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\n" ); document.write( "Taking the first case, showing that if p is a prime number of the form
\n" ); document.write( "6n + 1 , then there exists an m such that 3m + 1 = p.
\n" ); document.write( "Let p be such a prime number and n is a number such that 6n + 1 = p .
\n" ); document.write( "Certainly, 6n + 1 = 3(2n) + 1. So setting m = 2n , 3m + 1 = p.
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\n" ); document.write( "For the second case, we need to show that if p is a prime number of the form
\n" ); document.write( "3m + 1, then there exists an n such that 6n + 1 = p.
\n" ); document.write( "Let p be a prime number such that 3m + 1 = p.
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\n" ); document.write( "Suppose m is odd and greater than 2.
\n" ); document.write( "This means that 3m is odd, since an
\n" ); document.write( "odd number times an odd number is odd.
\n" ); document.write( "This means that 3m + 1 is even. But if
\n" ); document.write( "3m + 1 is even, 3m + 1 is not prime.
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\n" ); document.write( "Suppose m is even. Being even, there exists a number n
\n" ); document.write( "such that 2n = m. So 3m + 1 = 3(2n) + 1 = 6n + 1.\r
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