Potential sequences regarding the sums of perfect powers

[Dave Consiglio]

Hello,

I am considering the following potential sequences:

1. Numbers of the form c^p that can be expressed as the sum of two perfect powers (c^p  = a^m + b^n)

2. Numbers of the form c^ that cannot be expressed as the sum of two perfect powers (c^p != a^m + b^n for any a,b,c,m,n,p > 1)

Additional sequences could limit the bases and exponents (m != n != p or m >= n >= p >= 3).

For example, one possible sequnce is perfect powers of order 3 or larger that can be expressed as the sum of two perfect powers of order 3 or larger. That sequence begins:

128, 243, 512, 2048, 8192, 32768

where 128 = 2^7 = 4^3+ 2^6

243 = 3^5 = 3^3 + 6^3

etc.

These sequences are related to Beal's Conjecture (https://en.wikipedia.org/wiki/Beal_conjecture) that posits that no number of this form exists if a and b do not share a common factor. There is currently a $1M prize for a proof or counterexample.

I don't see any of these sequences in the OEIS. Do you all think they are worthy of inclusion? Any additional ideas to contribute? Please let me know!

Thanks,
Dave Consiglio

[M F Hasler]

On Thu, Jan 8, 2026 at 12:12 PM Dave Consiglio <dave...@gmail.com> wrote:

one possible sequnce is perfect powers of order 3 or larger that can be expressed as the sum of two perfect powers of order 3 or larger. That sequence begins:

128, 243, 512, 2048, 8192, 32768 

where 128 = 2^7 = 4^3+ 2^6

243 = 3^5 = 3^3 + 6^3

Seems to be a subsequence of A245713, first hit that I get for

https://oeis.org/search?q=%22Beal+conjecture%22

The third hit is A297867 : 3-powerful numbers that can be written as the sum of two coprime 3-powerful numbers

which starts with

3518958160000 = 3^4 * 29^3 * 89^3 + 7^3 * 11^3 * 167^3 = 2^7 * 5^4 * 353^3.

You get more hits searching for "Beal's conjecture". For example:

A261782

Powers C^z = A^x + B^y with positive integers A,B,C,x,y,z such that x,y,z > 2.

+10
4

16, 32, 64, 128, 243, 256, 512, 1024, 2048, 2744, 4096, 6561, 8192, 16384, 32768, ...

A264901

Sorted powers C^z = A^x + B^y with all positive integers and x,y,z > 2, with multiplicity.

-M.

[Dave Consiglio]

Hi M,

Thanks for these. I didn't think to search for the powers directly. This might render the sequences I proposed redundant.

Dave

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