Intermediate Mathematics-5
Let n be an integer greater than 3. Can 3 n − n 3 be composite for any n ?
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1 solution
Just as a note, n = 1 and n = 3 do not result in composite numbers. All other odd n do.
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Thank you, I have fixed the error.
I think that the more interesting question is for which n > 3 is 3 n − n 3 prime? A preliminary check shows that this is the case for n = 4 , 1 0 , 5 2 . I suspect that the list is infinite but it might be difficult to prove.
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@Jordan Cahn – Hahaha I had just found it on OEIS too and was about to add an edit to my comment, but you beat me to it. :) So it would appear as though it has not been determined yet if the list is infinite, (although as is often the case it is most likely to be infinite).
This is another interesting OEIS list. Note that it is speculated that a ( n ) > 0 for most n > 4 .
3 4 − 4 3 = 8 1 − 6 4 = 1 7 which is prime. You said any n
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Yes, it's prime for these n , but it is composite for all other n , so since it is composite for at least one n , the answer is "Yes".
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@Brian Charlesworth , wouldn’t be better if the question said for “some” n instead of any n , because if we found a counter example, the above statement would be false for any integer.
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@Hana Wehbi – I can see your point that "some" might be better than "any", but I think that for the present phrasing, if we can find one example of n where the expression is composite then the answer is legitimately "Yes". If the statement had been "Is 3 n − n 3 composite for all n ?", then finding a counterexample would lead to an answer of "No", but I don't think that this is equivalent to the present statement. We can leave it up to @EKENE FRANKLIN if they want to change "any" to "some" for sake of clarity.
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@Brian Charlesworth – Ok, makes sense. Thank you.
Yes. 3 n − n 3 for any odd n > 3 would result in an even number, which is composite.