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Question

How many numbers of form $a_0^{b_0}-b_0^{a_0}$ are a "nontrivial" sum of four such numbers $a_i^{b_i}-b_i^{a_i}$ ?

The "nontrivial" means: all unordered pairs $\{a_i,b_i\}$ are distinct, $a_i^{b_i}\ne b_i^{a_i}$ and $1 \lt a_i\lt b_i$.

This implies that such summands are positive (are in OEIS A045575), except $2^3-3^2 = -1$.

The only "nontrivial" examples I could find are:

$$\begin{align} (2^5-5^2) + (2^6-6^2) + (2^7-7^2) + (4^5-5^4) &= (3^6-6^3) \\ (2^8-8^2)+ (4^5-5^4) + (4^6-6^4) + (3^{10}-10^3) &= (4^8-8^4) \end{align}$$

Are these two the only such numbers?

For comparison, I suspect such sums with less than four summands do not exist, and that there are infinitely many such sums with more than four summands. With four summands exactly, I have only these two examples, hence this question.

Are there any other references (than ones listed in OEIS A045575) on problems related to $x^y-y^x$ numbers?


Background

These two numbers correspond to the following two Base-Exponent Invariant numbers:

$$\begin{array}{} 1464 &=& 2^5 + 2^6 + 2^7 + 4^5 + 6^3 &=& 5^2 + 6^2 + 7^2 + 5^4 + 3^6 \\ 68521 &=& 2^8 + 4^5 + 4^6 + 3^{10} + 8^4 &=& 8^2 + 5^4 + 6^4 + 10^3 + 4^8 \end{array}$$

That is, these numbers are a special case of the "Base-Exponent Invariant" numbers.

I call these "Order-$5$ Genus-$1$" Base-Exponent Invariant numbers $1464,68521\in G^{(5)}_1$.

In general, I have found only $14$ examples (see "short examples" in this answer) of "Order-$5$ Base-Exponent Invariant numbers". The largest known example is around $6\cdot 10^6$, while the next one, if it exists, is larger than $10^{16}$.


General near examples

I've searched for smallest "error" $e(n)$ such that "some elements plus the error" are a sum of the "other elements" from the "best" 5-subset of A045575 among "nontrivial" 5-subsets whose largest element is the $n$th nonzero term of A045575.

If $e(n)=0$ (and $n\ge 5$) then we have general examples(s) and $(n,0)$ is colored blue (or green if corresponding example is also "Genus-$1$"). If $e(n)=\pm 1$ we have a "near example" (colored red). Else, we have a black point $(n,\log e(n))$. For $n$ up to $100$, we have the log plot of errors:

enter image description here

Notice that for $n\gt 43$, we have no general examples, and for $n\gt 25$ we have no "Genus-$1$" examples (the examples I'm asking about in this question), so far.

It would seem that new examples are very large and rare or do not exist. However, notice the far right "near example" (red point) at $n=83$, which gives us hope

$$ (2^8-8^2) + (2^{16}-16^2) + (4^{16}-16^4) + (2^{32}-32^2) = (2^{33}-33^2) \color{red}{+1} $$

that maybe a large example could exist.

Do there exist any larger general examples, Genus-1 examples or near examples?

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    $\begingroup$ $2^3-3^2=-1$ is not the only exception. $2^4-4^2 = 0$. That should be also consider as trivial? $\endgroup$
    – jjagmath
    Sep 19, 2020 at 22:14
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    $\begingroup$ @jjagmath It is included as trivial: The "nontrivial" condition contains "$a_i^{b_i}\ne b_i^{a_i}$", implying summands that equal zero are not being considered. $\endgroup$
    – Vepir
    Sep 19, 2020 at 22:44

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