document.write( "Question 448561: Which of the following CANT be expressed as the sum of the squares of two integers? Is there any formula to solve such problems?
\n" ); document.write( "(a)13
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\n" ); document.write( "(e)34 \n" ); document.write( "
Algebra.Com's Answer #308654 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The other tutor didn't give a formula but there is one.\r\n" );
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document.write( "Euler proved that a positive integer N is expressible as the sum \r\n" );
document.write( "of the squares of two different integers if and only if in the \r\n" );
document.write( "prime factorization of N, every prime number of the form (4k+3),\r\n" );
document.write( "where k is a positive integer, occurs an even number of times.  \r\n" );
document.write( "Note that 0 is an even number, so if a prime of the form (4k+3),\r\n" );
document.write( "where k is a positive integer, doesn't occur at all in the prime \r\n" );
document.write( "factorization, then it is considered as occuring an even number \r\n" );
document.write( "of times.\r\n" );
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document.write( "The numbers of the form 4k+3 are \r\n" );
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document.write( "k    4k+3\r\n" );
document.write( "1      7\r\n" );
document.write( "2     11 \r\n" );
document.write( "3     15, not prime\r\n" );
document.write( "4     19 \r\n" );
document.write( "5     23\r\n" );
document.write( "6     27, not prime\r\n" );
document.write( "7     31\r\n" );
document.write( "8     35, not prime \r\n" );
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document.write( "So here is the list of primes of the form 4k+3 \r\n" );
document.write( "which are less than the numbers in your problem: \r\n" );
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document.write( "7,11,19,23,31\r\n" );
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document.write( "(a) 13\r\n" );
document.write( "The prime factorization of 13 is just 13 which contains \r\n" );
document.write( "all those in the list 0 times, and 0 is an even number, \r\n" );
document.write( "so 13 can be written as the sum of two squares of integers, \r\n" );
document.write( "2²+3²=4+9=13 \r\n" );
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document.write( "(b) 17\r\n" );
document.write( "The prime factorization of 17 is just 17 which contains \r\n" );
document.write( "all those in the list 0 times, and 0 is an even number, \r\n" );
document.write( "so 17 can be written as the sum of two squares of integers,\r\n" );
document.write( "1²+4²=1+16=17\r\n" );
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document.write( "(c) 21\r\n" );
document.write( "The prime factorization of 21 is 3*7 which contains \r\n" );
document.write( "7 one time, and 1 is an odd number, so 21 cannot\r\n" );
document.write( "be written as the sum of two squares of integers.\r\n" );
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document.write( "(d) 29\r\n" );
document.write( "The prime factorization of 29 is just 29 which contains \r\n" );
document.write( "all those in the list 0 times, and 0 is an even number, \r\n" );
document.write( "so 29 can be written as the sum of two squares of integers, \r\n" );
document.write( "2²+5²=4+25=29\r\n" );
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document.write( "(e) 34\r\n" );
document.write( "The prime factorization of 34 is just 2*17, but neither 2\r\n" );
document.write( "nor 17 are in the list of primes of the form 4k+3, so they \r\n" );
document.write( "don't matter.  So 34's prime factorization contains all \r\n" );
document.write( "those in the list 0 times, and 0 is an even number, so 34 \r\n" );
document.write( "can be written as the sum of two squares of integers, \r\n" );
document.write( "3²+5²=9+25=34\r\n" );
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document.write( "Edwin
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