Known solutions of the Fermat-Catalan equation $x^p+y^q=z^r$

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Numbers

solution

$x$, $y$, $z$, or $z^r$

$2^5+7^2=3^4$

$x$:

$2^5+7^2=3^4$

$y$:

$2^5+7^2=3^4$

$z$:

$2^5+7^2=3^4$

$z^r$:

$13^2+7^3=2^9$

$x$:

$13^2+7^3=2^9$

$y$:

$13^2+7^3=2^9$

$z$:

$13^2+7^3=2^9$

$z^r$:

512

$2^7+17^3=71^2$

$x$:

$2^7+17^3=71^2$

$y$:

$2^7+17^3=71^2$

$z$:

$2^7+17^3=71^2$

$z^r$:

5041

$3^5+11^4=122^2$

$x$:

$3^5+11^4=122^2$

$y$:

$3^5+11^4=122^2$

$z$:

122

$3^5+11^4=122^2$

$z^r$:

14884

$33^8+1549034^2=15613^3$

$x$:

$33^8+1549034^2=15613^3$

$y$:

1549034

$33^8+1549034^2=15613^3$

$z$:

15613

$33^8+1549034^2=15613^3$

$z^r$:

3805914951397

$1414^3+2213459^2=65^7$

$x$:

1414

$1414^3+2213459^2=65^7$

$y$:

2213459

$1414^3+2213459^2=65^7$

$z$:

$1414^3+2213459^2=65^7$

$z^r$:

4902227890625

$9262^3+15312283^2=113^7$

$x$:

9262

$9262^3+15312283^2=113^7$

$y$:

15312283

$9262^3+15312283^2=113^7$

$z$:

113

$9262^3+15312283^2=113^7$

$z^r$:

235260548044817

$17^7+76271^3=21063928^2$

$x$:

$17^7+76271^3=21063928^2$

$y$:

76271

$17^7+76271^3=21063928^2$

$z$:

21063928

$17^7+76271^3=21063928^2$

$z^r$:

443689062789184

$43^8+96222^3=30042907^2$

$x$:

$43^8+96222^3=30042907^2$

$y$:

96222

$43^8+96222^3=30042907^2$

$z$:

30042907

$43^8+96222^3=30042907^2$

$z^r$:

902576261010649

Definition

Known coprime positive-integer solutions of the Fermat-Catalan equation $x^p+y^q=z^r$ [1], with $x,y,z>1$, $\frac1p+\frac1q+\frac1r<1$ and $x^p<y^q$. The stored integers $x$, $y$, $z$ and $z^r$ are parts of one solution.

Parameters

solution

— sum (a perfect power $z^r$ from a coprime solution of $x^p+y^q=z^r$ with $x,y,z>1$ and $\frac1p+\frac1q+\frac1r<1$)

Comments

(1)

The summands are ordered by size, so $x^p<y^q$. Thus the solution with $z^r=512$ is written here as $13^2+7^3=2^9$, although some sources list the two summands in the other order.

(2)

The exponents $p$, $q$ and $r$ of each solution are shown in its label.

(3)

For any fixed positive exponents $p$, $q$ and $r$ with $\frac1p+\frac1q+\frac1r<1$, there are only finitely many coprime solutions [2].

(4)

The infinite family $1^p+2^3=3^2$ with $p>6$ has $x=1$ and gives the OEIS A214618 term $9$ [4]. It is represented by this comment rather than by rows, because $p$ is unbounded while the perfect-power triple is always $(1,8,9)$.

References

[1]

Adam S. Sikora, Fermat-Catalan and Tijdeman-Zagier Conjectures for Products, 2024. (arXiv)

[2]

Henri Darmon and Andrew Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, Bulletin of the London Mathematical Society 27 (1995), 513-543. (doi) (MR)

Links

[3]

Wikipedia: Fermat-Catalan conjecture

[4]

OEIS A214618

[5]

MathWorld: Fermat-Catalan Conjecture

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Data properties

Entries are of type: integer

How well the digits are known: exact

Table is complete: no (it holds the nine solutions with $x,y,z>1$ among the ten known coprime solutions listed in [1]; Sikora's 2024 search found no other coprime solution with $\min(p,q,r)\leq113$ and $x^p,y^q,z^r<M_{\min(p,q,r)}$, where $M_2=2^{71}$, $M_3=2^{80}$, $M_4=2^{100}$ and $M_d=2^{113}$ for $5\leq d\leq113$)

Sources of data: [1], [4], [3], [5]

How they were obtained:

Every stored value is an exact integer. The generator checks each identity, each coprimality condition and each exponent inequality with integer arithmetic. It also checks the stored sums against the OEIS A214618 b-file after removing its first term $9$, which comes from the excluded family $1^p+2^3=3^2$.