Find the largest positive integer $d$ with a property, that for any positive integer $n$ the value of expression $V(n) = n^4+11n^2-12$ is a multiple of $d$.
Could somebody give me a hint on how to start solving this problem?
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compute $ V(2) $ and see. – hamam_Abdallah Dec 17 '21 at 20:38
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How will that exact value help me? – badatmath Dec 17 '21 at 20:43
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The divisors of $ V(2) $ are the possible values of $ d$. – hamam_Abdallah Dec 17 '21 at 20:45
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I understand, but can I find out the largest possible value of $d$ without computing $V(n)$ for n > 2? – badatmath Dec 17 '21 at 20:48
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So we are looking for the greatest common divisor of $V(1), V(2), V(3), \ldots$. This must be a divisor of $V(2), V(3), \ldots$. If you compute a few of these smaller values, you can make a hypothesis on what $d$ is, and then prove it using modular arithmetic, number theory, maybe even induction not sure. – Alan Abraham Dec 17 '21 at 20:53
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The number $d$ certainly divides $V(2)$, so you can compute it and already find that there are only a finite number of possible $d$'s. – Kolja Dec 17 '21 at 20:55
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@hamam_Abdallah I don't think that claim is true. Just because $5$ is not a divisor of $V(5)$ doesn't mean that $6$ cannot be. We don't need to find the largest $d$ such that all integers below it divide $V(n)$. – Alan Abraham Dec 17 '21 at 20:55
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$$n^4+11n^2-12=(n^2-1)(n^2+12)$$ – hamam_Abdallah Dec 17 '21 at 21:16
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The same methods in the linked dupe apply here. – Bill Dubuque Dec 17 '21 at 21:25
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This question was improperly closed. The question linked looks similar but is actually not. – Mike Dec 17 '21 at 21:49
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@Mike Please read the prior comment. – Bill Dubuque Dec 17 '21 at 22:16
1 Answers
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Note that we may verify that for any $d$ we have
$$V(n+d)\equiv V(n)\ (\text{mod }d)$$
Thus, in order to check that a particular $d$ has the desired property, it is sufficient to check the integers $\{0,1,...,d-1\}$. As a hint on how to continue, note that $d$ divides $V(2)=48=2^4\cdot 3$ (as pointed out by @hamam_Abdallah). Can you take it from here?
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1@badatmath The expression basically means is that let's say you think a certain $d$ satisfies the required properties. Then you simply need to prove that $d|V(n)$ for all $n\in{0,1,\ldots d-1}$. – Alan Abraham Dec 17 '21 at 20:56