# Are (1,2) and (4,5) the only two consecutive pairs in A003592?

All odd numbers in A003592 are powers of $$5$$, so this is equivalent to finding all $$n \ge 0$$ such that $$5^n = 2^m+1$$ or $$2^m - 1$$ for some $$m \ge 0$$.

By quick brute force computation I cannot seem to find any consecutive pairs other than $$(1,2)$$ and $$(4,5)$$ for items under $$2^{32}$$. Furthermore, $$(125,128)$$ is the only pair that differs by $$3$$, but $$(25,32)$$ is the only pair that differs by $$7$$. I know these are more or less pretty "random" (in the sense that they don't have enough pattern to be worth attention), but I would like to know more about this distribution.

In particular, I would like a mathematical proof that $$(1,2)$$ and $$(4,5)$$ are the only pair that differs by one, and in general, I wonder if there is a function for an (approximate?) lower bound on the difference between consecutive terms of A003592.

For quick reference, here is a sorted list of differences below $$2^{32}$$: https://gist.github.com/SOF3/5c62561770b55fccc261beedd41c9e1f

Btw. A corollary of this conjecture states that $$(n,n,2n)$$ and $$(n,4n,5n)$$ are the only tuples taking only (duplicate) terms from A003592 that satisfy an $$(a,b,a+b)$$ condition. Otherwise, for $$a, b \ne 1$$, $$\gcd(a,b,a+b) = 1 \implies \gcd(a,b) = 1 \implies \text{(wlog) } 5 \mid a \wedge 2 \mid b \implies a + b \not\equiv 5 \pmod{10} \implies 2 \mid a + b \implies \gcd(b, a+b) = 2 \implies 2 \mid a \implies \mathbb F$$.

Context why I am interested in this question: I have some OCD where I try to make numbers in my life, e.g. number of steps I walk, number of syllables I talk, etc. be some number in A003592. When I am uncertain where I should count a certain step/syllable, I want the number to be in A003592 no matter it is counted or not.

• Catalan's conjecture, now Mihăilescu's theorem, tells us that the only consecutive perfect powers are $8,9$.
– lulu
Sep 8, 2020 at 15:35
• Great, that answers the specific case for $(1,2)$ and $(4,5)$. But is there any relevant theorem regarding the lower bound of consecutive differences?
– SOFe
Sep 9, 2020 at 4:15
• A lower bound can be obtained from "linear forms in logarithms": see the first answer to this question, where they speak of powers of $2$ and $3$ but the method works equally well for powers of $2$ and $5$. Sep 9, 2020 at 4:53