Does anyone have an idea on how to prove this?
"There is no strictly monotonically increasing arithmetic sequence in which all elements are primes."
Any help appreciated! Thanks:)
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1$\begingroup$ Well, write what it means to be a monotonically increasing arithmetic sequence. $\endgroup$ – Thomas Andrews Sep 17 '13 at 17:16
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If $p$ is a polynomial with integer coefficients, and $k$ divides $p(n)$, then $k$ divides $p(mk+n)$ for any $m$.
In particular, this applies to $p$ a linear polynomial. Now note that terms in arithmetic progression are the consecutive values of a linear polynomial with integer coefficients: $a,a+b,a+2b,\dots$ are $p(0),p(1),\dots$ for $p(x)=a+xb$.
answered Sep 17 '13 at 17:16
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$\begingroup$ That sounds good, but how do I know that there is an element in the sequence that's equal to p(mk + n)? $\endgroup$ – nitrogenhurricane Sep 17 '13 at 17:21
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$\begingroup$ The $(mk+n)$-th member of the sequence is is $p(mk+n)$. $\endgroup$ – Andrés E. Caicedo Sep 17 '13 at 17:22
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$\begingroup$ In case you are confused about this: For any $N$, no matter how large, there is a strictly increasing arithmetic progression of length $N$ consisting solely of primes. The question is about infinite sequences, so no matter how large $k,n,m$ are, the sequence has an $(mk+n)$-th term. $\endgroup$ – Andrés E. Caicedo Sep 17 '13 at 17:25
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$\begingroup$ Okay, it's clear now thanks, I just couldn't put the existence of element and that k divides it together:) Thanks! $\endgroup$ – nitrogenhurricane Sep 17 '13 at 17:26
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Suppose the first term in the sequence is prime $p$. Then the $(p+1)$th term in the sequence is $p + pn$ for some positive integer $n$, since by assumption your sequence must be all integer-valued if it's all prime. Can this term $p + pn$ be prime?
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$\begingroup$ @nitrogenhurricane, in this answer the difference between two elements is $n$, not $p$. $\endgroup$ – Antonio Vargas Sep 17 '13 at 17:54
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Consider the arithmetic sequence $a,a+d,a+2d,a+3d,\dots$. The sequence is increasing, so for some $k$ we have $a+kd\gt 1$.
Now consider $(a+kd)+ (a+kd)d$. This is in the sequence, for $$(a+kd)+(a+kd)d= a+(k+a+kd)d.$$ But $(a+kd)+(a+kd)d$ is not prime, since it is clearly divisible by $a+kd$, and greater than $a+kd$.
answered Sep 17 '13 at 17:22
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$\begingroup$ Thanks for your answer, but as Andres was faster I think, so it's fair to accept his. Please anyone correct me if I'm wrong since I'm a newbie here. $\endgroup$ – nitrogenhurricane Sep 17 '13 at 17:27
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$\begingroup$ In my opinion the solution by user2566092 is much cleaner and better. Mine is essentially the same, but hides the simplicity of the idea. So I think you should not accept mine. $\endgroup$ – André Nicolas Sep 17 '13 at 17:31
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$\begingroup$ You should accept whichever answer you think best answers the question. It's okay for this to be a late answer (it's even okay to change your mind) $\endgroup$ – Ben Millwood Sep 17 '13 at 17:32
