Wikipedia:Reference desk/Archives/Mathematics/2023 November 8

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November 8

Consecutive odd powers between squares

Call a perfect power (OEIS A001597) odd if it's not a square, in other words if its exponent is odd >1. So 8, 27, 32, ... , but do not include 64, 729, 1024, . Are there an infinite number of pairs of odd powers between consecutive squares? Examples of such pairs are {27, 32} between 25 and 36, {125, 128} between 121 and 144, {2187, 2197} between 2116 and 2209. I'm not really expecting much here since such questions like this tend to be difficult; see for example Catalan's conjecture and the related Pillai's conjecture. But I don't think there's harm in asking. --RDBury (talk) 08:13, 8 November 2023 (UTC)

PS. According to my calculations the next pair is {6434856 (=186³), 6436343 (=23⁵)}. --RDBury (talk) 08:37, 8 November 2023 (UTC)

I don't have anything to solve the actual problem of infinitude, but there is a function that makes finding examples easier. The function $\lfloor \lceil x^{k_{1}/2}\rceil ^{2/k_{2}}\rfloor -\lfloor \lfloor x^{k_{1}/2}\rfloor ^{2/k_{2}}\rfloor$ tells you, given an input $x$, how many $k_{2}$-th powers occupy the same niche between squares that $x^{k_{1}}$ does. So for example, if $k_{1}=5$, then $x=2$ yields $1$, since there is one cube, $27$, in the same niche $(25,36]$ that $32$ occupies. Naturally, if $x$ is a $k_{2}$-th power, then this does not correspond to a distinct pair of odd powers, since the niche is occupied by the single number $x^{k_{1}k_{2}}$. Also, if $\lceil x^{k_{1}/2}\rceil$ is a $k_{2}$-th power, then there is $k_{2}$-th power occupying the same niche, but it is the higher square defining the niche itself. For all other $x$ for which this equation is $1$ (of course, assuming that $k_{1},k_{2}$ are odd), one gets a distinct pair of odd powers within the niche. GalacticShoe (talk) 17:32, 8 November 2023 (UTC)

That's a lot more efficient that the method I was using. I used your formula to get additional pairs {312079650687, 312079600999}, {328080696273, 328080401001}, {11305786504384, 11305787424768}, {62854898176000, 62854912109375}, {79723529268319, 79723537443243}, {4550858431781696, 4550858390629024}, and that's only with k₁=3, k₂=5. So I'm beginning to suspect that the answer is that there are an infinite number of pairs. --RDBury (talk) 07:17, 9 November 2023 (UTC)

One might even suspect that the answer is still positive when the question confines the odd powers to powers of 3 and 5.  --Lambiam 09:50, 9 November 2023 (UTC)

"Pairs of odd powers between consecutive squares" as a section title would be a better description of the topic.  --Lambiam 09:21, 9 November 2023 (UTC)

A preposition that implies the proposition in the question:

For every pair of positive real numbers $a$ and $b$, the inequation $2^{n}<\{9^{p}a,25^{q}b\}<2^{n{+}1}$ has infinitely many solutions.

(Taking $a=3$ and $b=5$ turns the inhabitants of the middle pair into odd powers.) Would this proposition then follow from the following one:

The set of numbers of the form $p\log 9-q\log 25$ is dense in the reals.

?  --Lambiam 10:10, 9 November 2023 (UTC)

I now think this is not going to help.  --Lambiam 21:26, 9 November 2023 (UTC)

OEIS:A117594 is related. PrimeHunter (talk) 12:38, 9 November 2023 (UTC)

OEIS:A117934 is more closely related and has all the pairs listed above. Some of the crossrefs in that entry are related as well. Given that 180 pairs have been computed it seems unlikely that the number is finite. But I assume that if there were a known proof of this then the OEIS would have a link to it. --RDBury (talk) 13:39, 9 November 2023 (UTC)

I looked for a pattern to generate solutions. (10^(6*n+3)+6*10^n)^5 and (10^(10*n+5)+10^(5*n+3)+2)^3 are close. Often a square squeezes in between them but there are 3693 solutions for n up to 10000. It appears to remain around 37% so I guess there are infinitely many solutions of this form. PrimeHunter (talk) 22:37, 9 November 2023 (UTC)

This can be generalized by using

$(5^{3}p^{3}q^{6}+3pq)^{5},(5^{5}p^{5}q^{10}+5^{3}p^{3}q^{5}+p)^{3}.$

The pattern above is obtained by substituting $p:=2,q:=10^{n}.$  --Lambiam 01:21, 10 November 2023 (UTC)

Nice. I think there is a further slight generalization/variation: Let rp=5s² where s→∞ and p ≤ 4. Then take the pair (r³+3s)⁵=N₅ and (r⁵+5r²s+p)³=N₃, It's clear that N₅, N₃ ∼ s³⁰, where ∼ means asymptotically equal. According to my calculations N₃ - N₅ ∼ 5r⁶s³ and √N₃ - √N₅ ∼ p/2√5 < 1. This means that the interval [√N₃, √N₅] has a positive "probability" of not containing an integer. I wonder if this technique can be used to generate more examples with exponents higher than 5. I gather none are known with exponent 11. --RDBury (talk) 05:32, 10 November 2023 (UTC)

PS. I think the smallest pair produced by this scheme is 91134⁵ = 6286402185474975604883424, 184558501³=6286402185476096945425501. This is with s=3, p=1, r=45 and corresponds to entries 49 and 50 on the list given in OEIS here. It appears that the pairs produced this way account for a very small percentage of all pairs. --RDBury (talk) 09:13, 10 November 2023 (UTC)

PPS. I just realized there is nothing really preventing s from being negative. N₃ < N₅ in this case but everything else works the same way. Taking s=-2, p=1, m=20 gives a smaller pair, {32645304179257588001, 32645304181811832224}, which are entries 31 and 32 in the list linked to above. --RDBury (talk) 09:55, 10 November 2023 (UTC)