if one of the integers $m,n$ is $1$ it does not seem too difficult to find examples of odd primes satisfying: $$|p^m-q^n| = 2$$ so suppose $\min(m,n)>1$, and call (just for the purpose of this question) primes satisfying the condition mentioned complementary. what can be asserted about the set of complementary primes? is such complementarity rare, or quite common?
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1How far have you looked for examples? Do the pairs $(9,11),(23,25),(125,127)$ fit into your initial question? (It is fairly frequent that there is a prime within distance $2$ on one side or the other of the square of a prime...) – abiessu Dec 23 '13 at 16:57
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It is commomplace to see "|" used where "\mid" should appear, but this was the opposite error. I changed it. – Michael Hardy Dec 23 '13 at 17:00
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@MichaelHardy: what is the usual usage of "\mid"? – abiessu Dec 23 '13 at 17:01
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\mid is used in $\Pr(A\mid B)$ or ${f(x)\mid x\in A}$ or the like. \mid provides spacing before it and after it, which would be absent if one were to write (as many here do) $\Pr(A|B)$ or ${f(x)|x\in A}$. You wrote $\mid p^m-q^n\mid=2$. I changed it to $|p^m-q^n|=2$. Notice that the difference in appearance is conspicuous. – Michael Hardy Dec 23 '13 at 17:07
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@abiessu yes I noticed that from a few examples. but not so obvious with the $m,n \gt 1$ condition. I haven't searched thoroughly - if there are any obvious examples they will surely be presented. but the question is of interest, e.g. in relation to the prime pairs question ($m=n=1$) (which, if it hasan't been shown yet, nevertheless, like the well-known Goldbach and Riemann conjectures, nearly everyone except professional contrarians or sceptics, thinks very probable) – David Holden Dec 23 '13 at 17:54
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@Michael thanks for that. although a slow learner I am very interested in questions of mathematical orthography. I am new to mathjax, and you can perhaps imagine what a wonderful liberation it is to be able to construct reasonable notation in this way. – David Holden Dec 23 '13 at 17:56
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abiessu $3^3+2 =29; 5^3+2 =127$ – David Holden Dec 23 '13 at 18:09
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With the edit to your question, my examples are no longer relevant; the way it read initially, it seemed that you hadn't found examples for the case where exactly one of $m,n$ is $1$... – abiessu Dec 23 '13 at 18:26
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no problem with examples. it all helps develop the creative loam of the subconscious imagination. Dietrich's reference shows the gateway to answering this question, though as yet I lack the analytical equipment to enter through it. – David Holden Dec 23 '13 at 18:31
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The Diophantine equation $p^m-q^n=c$ for given integers $p,q,c$ is called Pillai's equation. Here one searches for positive integer solutions $m,n$. There are many conjectures (and some results) on this equation. For example, Pillai conjectured that $3^m-2^n=c$ has at most one solution in positive integers for $|c|>13$. This was proved by Stroecker and Tijdeman in 1982. For more details see http://www.math.ubc.ca/~bennett/B-Pillai.pdf.
What does this mean for our question, if $m,n>1$ are given ? As far as I can see, there are not many primes $p,q$ (I suppose that you want distinct primes) satisfying Pillai's equation with given exponents.
Dietrich Burde
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