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Let $A$ be an infinite set of positive integers. For any two $a,b\in A$, $a\neq b$, at least one of the numbers $a^b+2$ and $a^b-2$ are also in $A$. Must $A$ contain a composite number?

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  • $\begingroup$ what is a composite number? $\endgroup$
    – Paul
    Nov 27, 2014 at 7:36
  • $\begingroup$ a composite number is an integer larger than one, which is not prime. $\endgroup$
    – sranthrop
    Nov 27, 2014 at 7:43
  • $\begingroup$ Note that if $A$ does not contain a composite number then all $a$ must be odd. $\endgroup$
    – Arian
    Nov 27, 2014 at 8:58
  • $\begingroup$ @Arian,if I understood the problem correctly,all b must also be odd(since a and b play a symmetric role). $\endgroup$
    – rah4927
    Nov 28, 2014 at 4:05
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    $\begingroup$ It is interesting to note that this is false in the finite case, let $A=\{1,3\}$ $\endgroup$
    – gregkow
    Nov 29, 2014 at 22:03

1 Answer 1

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The answer is yes.

Take four odd numbers $a<b<c<d$ from the set, all greater than $3$, such that $c=b^a\pm2$ and $d=c^b\pm2$.

Case 1: $c=b^a\pm2$ and $d=c^b\pm2$ (with equal signs).

Then $c=b^a\pm2\equiv b\pm2 \pmod3$ and $d=c^b\pm2\equiv c\pm2 \pmod3$. One of $b,c,d$ is divisible by $3$.

Case 2: $c=b^a\pm2$ and $d=c^b\mp2$ (with opposite signs). If $b$ is a prime then, by Fermat's theorem, $$ d = c^b \mp2 \equiv c\mp2 = b^a \equiv 0 \pmod{b} $$ so $d$ is divisible by $b$. (Obviously $d$ is greater than $b$.)

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