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Fermat-Catalan Conjecture


The conjecture that there are only finitely many triples of relatively prime integer powers x^p, y^q, z^r for which

 x^p+y^q=z^r
(1)

with

 1/p+1/q+1/r<1.
(2)

Darmon and Merel (1997) have shown that there are no relatively prime solutions (x,x,3) with x>=3. Ten solutions are known,

 1^p+2^3=3^2
(3)

for p>6, and

2^5+7^2=3^4
(4)
7^3+13^2=2^9
(5)
2^7+17^3=71^2
(6)
3^5+11^4=122^2
(7)
17^7+76271^3=21063928^2
(8)
1414^3+2213459^2=65^7
(9)
9262^3+15312283^2=113^7
(10)
43^8+96222^3=30042907^2
(11)
33^8+1549034^2=15613^3
(12)

(Mauldin 1997).

The following table summarizes known solutions (Poonen et al. 2005). Any remaining solutions would satisfy the Tijdeman-Zagier conjecture, also known popularly as Beal's conjecture (Elkies 2007).

exponents (p,q,r)reference
(2, 3, 7)Poonen et al. (2005)
(n,n,n)Wiles
(2, 3, 8), (2, 3, 9), (2, 4, 5),Bruin (2004)
(2, 4, 6), (3, 3, 4), (3, 3, 5)
(2, 4, 7)Ghioca
(2,n,n), (3,n,n)Darmon-Merel
(2n,2n,5)Bennett
(2,4,n)Bennett-Skinner

It is not known if the analogous conjecture for x, y, and z Gaussian integers holds. Known solutions include

(8+5i)^2+(5+3i)^3=(1+2i)^7
(13)
(20+9i)^2+(1+8i)^3=(1+i)^(15)
(14)

(E. Pegg Jr., pers. comm., March 30, 2002).


See also

Catalan's Conjecture, Fermat's Last Theorem, Fermat's Sandwich Theorem, Unit Fraction

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References

Bruin, N. "Visualising Sha[2] in Abelian Surfaces." Math. Comput. 73, 1459--1476, 2004.Darmon, H. and Granville, A. "On the Equations z^m=F(x,y) and Ax^p+By^q=cZ^r." Bull. London Math. Soc. 27, 513-543, 1995.Darmon, H. and Merel, L. "Winding Quotients and Some Variants of Fermat's Last Theorem." J. reine angew. Math. 490, 81-100, 1997.Elkies, N. "The ABCs of Number Theory." Harvard Math. Rev. 1, 64-76, 2007.Mauldin, R. D. "A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem." Not. Amer. Math. Soc. 44, 1436-1437, 1997.Poonen, B.; Schaefer, E. F.; and Stoll, M. "Twists of X(7) and Primitive Solutions to x^2+y^3=z^7." 10 Aug 2005. http://arxiv.org/abs/math/0508174.

Referenced on Wolfram|Alpha

Fermat-Catalan Conjecture

Cite this as:

Weisstein, Eric W. "Fermat-Catalan Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Fermat-CatalanConjecture.html

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