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| Primes (Posted on 2003-09-09) |
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If x and (x² + 8) are both primes, then prove that (x³ +16) is also a prime.
First thoughts...
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 | Comment 1 of 9
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Note that 2 is the only even prime and 2²+8 is not prime, therefore, x must be odd.
Note also that x can't end in 5 because then it is not prime (unless x=5, but 5²+8 = 33 which is not prime). Therefore x must end in 1, 3, 7, or 9.
If x ends in 9 then x³ ends in 9, and x³+16 ends in 5 (which can't be prime--it is divisible by 5). Therefore x ends in 1, 3, or 7.
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