document.write( "Question 269079: What is the only prime number that is the sum and difference of two primes? \n" ); document.write( "

Algebra.Com's Answer #854063 by n2(79) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "What is the only prime number that is the sum and difference of two primes?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "It the prime number 'p' is the sum of two primes p1 and p2\r\n" );
document.write( "\r\n" );
document.write( "    p = p1 + p2,    (1)\r\n" );
document.write( "\r\n" );
document.write( "then it is clear from the parity considerations that one of these numbers, p1 and p2,\r\n" );
document.write( "must be '2'.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Also, it is clear that if the prime number 'p' is the difference of two primes p1 and p2\r\n" );
document.write( "\r\n" );
document.write( "    p = p3 - p4,    (2)\r\n" );
document.write( "\r\n" );
document.write( "then one of these numbers, namely p4, must be '2'.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus we should have\r\n" );
document.write( "\r\n" );
document.write( "    p = p1 + 2,    (3)\r\n" );
document.write( "\r\n" );
document.write( "    p = p3 - 2.    (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From these equations, we see that the difference  p3-p1  is  4  (taking the difference of these equations).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "One possible example of such triple is   3, 5 and 7.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Indeed, 2 + 3 = 5: the sum        of prime numbers 2 and 3 is a prime number 5,  and\r\n" );
document.write( "        7 - 2 = 5: the difference of prime numbers 7 and 2 is a prime number 5.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To prove the uniqueness, notice that from equations (3) and (4), the triple must be\r\n" );
document.write( "\r\n" );
document.write( "        p1,  p2 =   and p3\r\n" );
document.write( "\r\n" );
document.write( "with the difference between the adjacent terms of 2, so, p1, p2 and p3 should be three consecutive odd numbers.\r\n" );
document.write( "\r\n" );
document.write( "Hence, one of p1, p2 or p3 must be multiple of '3', and it makes necessary p1 = 3 (the minimal of the terms is 3).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It means that the triple (3, 5, 7) is the unique triple of this kind.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus, prime numbers (2, 3, 5) are the UNIQWUE triple satisfying the problem's conditions.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Prime number 5 is the ONLY prime number, which is the sum and the difference\r\n" );
document.write( "of two primes, at the same time.    ANSWER\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );