Is it true that if $x,y,m,\delta$ are integers, $\gcd(x,y)=1$, $m\ge2$, $\delta\ge1$, then $$|x^m-y^{m+\delta}|\ge\delta?$$
Any proofs or references will be most welcome.
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Can you specify x,y, m and $ \delta $ ? In which set are they? Natural numbers, real numbers etc. ? Is there some relation between them? – Imago Dec 24 '15 at 21:48
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$|8^2-4^{2+1}|<1$. – Gerry Myerson Dec 24 '15 at 21:48
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Sorry had forgotten the condition that $(x,y)$ are relatively prime. – Q_p Dec 24 '15 at 21:54
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2@Isaac. Please consider that it might be a bit rude to radically change the question after it's already been answered. – Dec 24 '15 at 21:54
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@User, noted with thanks, and any inconveniences caused are sincerely regretted. – Q_p Dec 24 '15 at 21:59
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1Isaac, the thing to do is to get some easy computer language, people seem to like Python for example. Then you can run your own simulations before asking any questions. https://www.python.org/ I use C++ , I also have Sage and gp-Pari, all of which were free. – Will Jagy Dec 24 '15 at 22:22
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Just out of curiosity, what made you think of this particular statement? Is there any particular reason you thought it might hold? – A.P. Dec 24 '15 at 22:47
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1You might be interested in looking up the abc conjecture. – Gerry Myerson Dec 25 '15 at 05:19
