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I'm planning a challenge over on Code Golf.SE about integers $a, b, c \ge 0$ such that

$$a^n + b^n = c^n \pm 1$$

for a given integer $n > 2$. However, I'm interested in whether any non-trivial solutions to this exist for a given $n$. Here, I'm defining "non-trivial" solutions as triples $a, b, c$ such all three are unique and non-zero (i.e. to avoid $(a, 1, a)$ and $(a, 0, a)$, and related triples).

I've found this question which asks a related (and broader) question about the existence of such triples, and the accepted answer states

I think that if $n\ge5$ (and assuming the ABCD conjecture), then for any $k$, the equation $$ a^n + b^n - c^n = k $$ has only finitely many solutions $a,b,c\in\mathbb{Z}$ with $|a|,|b|,|c|$ distinct and non-zero.

However, this doesn't fully state whether there are a non-zero number of distinct, non-zero solutions.

This is a program which attempts to find such triples, with $0 \le a, b, c \le 100$, given an input $n$, but so far it hasn't found any for either $n = 4$ or $n = 5$, and it times out if you increase the upper limit by any significant amount.

Therefore, my question is:

  • Can it be shown that, for all integers $n > 2$, the equation $a^n + b^n = c^n \pm 1$ has at least 1 non-trivial solution, for $a, b, c \ge 0$?
  • If not, does expanding the range for $a, b, c$ to $\mathbb{Z}$ affect or change this?
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    $\begingroup$ What kind of weird messed-up language is your example written in? We're not on CodeGolf here, readability isn't considered a defect. $\endgroup$ Commented Nov 26, 2020 at 17:11
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    $\begingroup$ For $n = 4$ there are no solutions with $a, b \le 10^4$. I strongly suspect there are no solutions at all. (The folks at CodeGolf might be a bit unhappy if you challenge them to find an object which doesn't exist.) $\endgroup$ Commented Nov 26, 2020 at 18:13
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    $\begingroup$ Did you follow the links at that older question, card? In particular, the link to Noam Elkies' computations? $\endgroup$ Commented Nov 26, 2020 at 22:24
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    $\begingroup$ I think it's worth digging in, to get some idea of how unlikely one is to find any non-trivial examples of $a^n+b^n=c^n\pm1$ for $n\ge4$. $\endgroup$ Commented Nov 26, 2020 at 22:42
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    $\begingroup$ @JohnD.Cook The first line of the linked answer: “ A 4-variable version of the infamous ABC Conjecture says the following:” $\endgroup$ Commented Nov 29, 2020 at 13:39

2 Answers 2

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[EDITED] It is likely that there are no solutions at all for $n \ge 4$. For $n \ge 5$ a solution would be a counterexample to the Lander, Parkin, and Selfridge conjecture. The best FLT "near miss" that I know of is $13^5 + 16^5 = 17^5 + 12$.

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    $\begingroup$ "It is likely that there are no solutions at all." For $n \geq 4$, that is. For $10^3+9^3-12^3=1000+729-1728=1$. $\endgroup$ Commented Nov 29, 2020 at 2:39
  • $\begingroup$ Yes, I meant for $n \ge 4$. As Zhi-Wei Sun noted, there are infinitely many solutions for $n=3$. Editing. $\endgroup$ Commented Nov 30, 2020 at 6:17
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In a message "A conjecture related to Fermat's Last Theorem" sent to Number Theory List on Sep. 26, 2015, I wrote the following:

In 1936 K. Mahler discovered that $$(9t^3+1)^3 + (9t^4)^3 - (9t^4+3t)^3 = 1.$$ Clearly, $$|1^n+1^n-2^n| = 2^n-2\ \mbox{for every}\ n = 4,5,6,\ldots$$ and $$13^5+16^5-17^5 = 371293+1048576-1419857 = 12 < 2^5-2.$$

Here I report my following conjecture which can be viewed as a further refinement of Fermat's Last Theorem.

CONJECTURE (Sept. 24-25, 2015). (i) For any integers $n > 3$ and $x,y,z > 0$ with $\{x,y\}\not= \{1,z\}$, we have $$|x^n+y^n-z^n|\ge2^n-2,$$

unless $n = 5$, $\{x,y\} = \{13,16\}$ and $z = 17$.

(ii) For any integers $n > 3$ and $x,y,z > 0$ with $z\not\in\{x,y\}$, there is a prime $p$ with $$x^n+y^n < p < z^n\ \ \mbox{or}\ \ z^n < p < x^n+y^n, $$

unless $n = 5$, $\{x,y\} = \{13,16\}$ and $z = 17$.

(iii) For any integers $n > 3$, $x > y \ge0$ and $z > 0$ with $x\not=z$, there always exists a prime $p$ with
$$x^n-y^n < p < z^n\ \ \mbox{or}\ \ z^n < p < x^n-y^n. $$

I have checked this new conjecture via Mathematica. For example, I have verified part (i) of the conjecture for $n = 4,\ldots,10$ and $x,y,z=1,\ldots,1700$.

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