document.write( "Question 1185809: What is the fourth positive integer N such that the product of any two increased by 1 of the numbers 3,8,21,N is a square
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Algebra.Com's Answer #816688 by Edwin McCravy(20056)\"\" \"About 
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document.write( "The products of two numbers among 3,8,21 are 24,63,168.\r\n" );
document.write( "None of those products increased by 3,8, or 21 are squares.\r\n" );
document.write( "So those products 24,63,168 must depend upon adding N to them to\r\n" );
document.write( "give a square sum. Therefore, if there is any solution to the problem:\r\n" );
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document.write( "24+N = a2 for some positive integer a.\r\n" );
document.write( "63+N = b2 for some positive integer b.\r\n" );
document.write( "168+N = c2 for some positive integer c.\r\n" );
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document.write( "Thus N = a2-24 = b2-63 = c2-168\r\n" );
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document.write( "That only has one solution, a=5, b=8, and c=13, which makes N=1\r\n" );
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document.write( "However, that means the product 21N is 21. But, no matter which of these:\r\n" );
document.write( "3,8,21,1 we add to the product 21 gives a square for a sum.\r\n" );
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document.write( "Thus there is no possible positive integer N meeting the conditions\r\n" );
document.write( "that the product of any two increased by 1 of the numbers 3,8,21,N is a square.\r\n" );
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document.write( "This proves there is no solution. Could there be a typo, perhaps in the number\r\n" );
document.write( "21?\r\n" );
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document.write( "Edwin
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