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Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval $\small f_n(+1).. f_n(-1) $ or $$ {3^n+1 \over 2^n+1} \le x \le {3^n-1 \over 2^n-1} $$ an x can be integer (In the original version of this question I asked for a "crossing of an integer value" in $\small f_n(+1) \ldots f_n(-1) $ which seemed to be somehow obfuscating).

Obviously this interval is roughly centered around the noninteger number $ {3^n \over 2^n}$ and the question can be reformulated whether for all $n \gt 1$ $$ \left\lfloor {3^n \over 2^n} \right\rfloor \lt {3^n+1 \over 2^n+1} \le {3^n \over 2^n} \le {3^n-1 \over 2^n-1} \lt \left\lceil {3^n \over 2^n} \right\rceil $$ Or one could ask whether in the interval $\small f_n(1)<x=f_n(j)<f_n(-1) $ there is an integer x for some j where $ -1 \le j \le 1 $.

Empirically we'll find an integer only for $\small n \in \{1,4,7\} $ and moreover, such an integer for some negative j only at n=1 .

It's worth to note that $${3^n+ \frac12 (\frac43)^n \over 2^n+\frac12 (\frac43)^n} \ldots {3^n- \frac12 (\frac43)^n \over 2^n-\frac12 (\frac43)^n } $$ defines an interval which increases with $\small n \ge 1$ from width 0.375 up to $\small 1 $ (not including). Thus it can contain at most one integer number and the interval here in question $\small f(1) \ldots f(-1) $ or $ {3^n+ 1 \over 2^n+1} \ldots {3^n- 1 \over 2^n-1} $ is always enclosed within.

The problem (to prove this for n>7)

  • is related to the disproof of the so-called "1-cycles" in the collatz-problem (the proof of that Collatz-detail due to Ray Steiner in 1977 is based on the proof, that $\small g_n(k)_{for k>1} \notin \{2^m\} $ for n>1 which is a much sharper requirement and could be solved using results of transcendental number theory)
  • is also related to a detail in the Waring-problem which is unsolved due to Eric Weissstein at mathworld-wolfram.com
  • and has also a relation to the problem of z-numbers which were introduced by Kurt Mahler to solve a problem which was formulated quite similar.

so I don't expect a solution here.

But that problem "took me" quite a long time in the last years and I'm still curious in its relevant aspects, and I was just re-reading some of my older notes on it. So my question is, whether I can find more interesting discussions around this problem, may be in other formulations or with other focus just to possibly widen my horizon here (there is no nice general convention how to use numerical expressions in search-strings, so I ask here with the more verbal environment).

P.s. perhaps there might be some more appropriate tag for this question?

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I don't know what you mean by "integer crossings". –  Gerry Myerson Feb 13 '12 at 3:45
@Gerry: (improved my previous comment): that in an interval $\small f_n(1) \ldots f_n(-1) $ occurs one integer –  Gottfried Helms Feb 13 '12 at 7:59
So - this whole thing is just asking whether there's an integer between $(3^n+1)/(2^n+1)$ and $(3^n-1)/(2^n-1)$? –  Gerry Myerson Feb 13 '12 at 11:56
:-) Yes. This problem seems to be present under various different foci. I noted above three of those which I've seen so far and hope to find further hints where this has been discussed perhaps in other disguise or with other focus - hopefully there is some broader view, which sometimes helps to improve the understanding of a specific problem –  Gottfried Helms Feb 13 '12 at 12:48
Intresting question. Poorly stated somewhat. Intresting links on the other hand. Conclusion +1 :) –  mick Oct 11 '12 at 16:40

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