document.write( "Question 754348: Cube of a prime plus square(4 digit number) \n" ); document.write( "
Algebra.Com's Answer #459039 by Edwin McCravy(20081)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "There are mamy many possibilities.\r\n" );
document.write( "I'll get just the smallest and the largest possibile solutions.\r\n" );
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document.write( "The smallest prime is 2\r\n" );
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document.write( "It's cube is 2³ or 8\r\n" );
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document.write( "The smallest number we can add to 8 to get a 4-digit number is\r\n" );
document.write( "1000-8=992.  But 992 is not a square, so we want to add the smallest\r\n" );
document.write( "square bigger than 992. To do that we get its square root which\r\n" );
document.write( "is about 31.496. Then the next whole number bigger than that,\r\n" );
document.write( "which is 32, is such that if we square it and add it to 8, it\r\n" );
document.write( "will be the cube of a prime number plus a square.\r\n" );
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document.write( "So we add \r\n" );
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document.write( "2³ + 32³ = 8 + 1024 = 1032\r\n" );
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document.write( "That's the smallest possible solution, 1032.\r\n" );
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document.write( "Let's start with the largest prime whose cube is a \r\n" );
document.write( "4 digit number.  That's the prime number 19, because \r\n" );
document.write( "19³ = 6859, a four-digit number whereas the next prime \r\n" );
document.write( "23³ = 12167, which is a five-digit number. \r\n" );
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document.write( "It's cube is 19³ or 6859\r\n" );
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document.write( "The largest number we can add to 6859 to get a 4-digit number is\r\n" );
document.write( "9999-6859=3140.  But 3140 is not a square, so we want to add the largest\r\n" );
document.write( "square smaller than 3140. To do that we get its square root which\r\n" );
document.write( "is about 56.036. Then the largest whole number smaller than that,\r\n" );
document.write( "which is 56, is such that if we square it and add it to 6859, it\r\n" );
document.write( "will be the cube of a prime number plus a square.\r\n" );
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document.write( "So we add 19³ + 32³ = 8 + 1024 = 1032\r\n" );
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document.write( "19³ + 56² = 9995\r\n" );
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document.write( "That's the largest possible solution.\r\n" );
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document.write( "Edwin
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