6n^3 + 3 is not a perfect 6th power.

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SUMMARY

The expression 6n^3 + 3 cannot be a perfect sixth power for any natural number n. By analyzing the expression modulo 7, it is established that 6n^3 + 3 results in values of 2, 3, or 4. In contrast, sixth powers of integers yield results of either 0 or 1 modulo 7. Therefore, the conclusion is definitive: 6n^3 + 3 is not a perfect sixth power.

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Prove that the number 6n3 + 3 cannot be a perfect sixth power of
an integer for 'any natural number n.​3
 
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Re: 6n^3+3 is not a perfect 6th power.

If we consider modulo 7, then 6n^3 + 7 is 2, 3 or 4 but 6-th powers of any integer are 0 or 1 modulo 7. Hence, QED.
 

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