# The generalised Fermat equation x2 + y3 = z15

@article{Siksek2013TheGF,
title={The generalised Fermat equation x2 + y3 = z15},
author={Samir Siksek and Michael Stoll},
journal={Archiv der Mathematik},
year={2013},
volume={102},
pages={411-421}
}
• Published 17 September 2013
• Mathematics
• Archiv der Mathematik
We determine the set of primitive integral solutions to the generalised Fermat equation x2 + y3 = z15. As expected, the only solutions are the trivial ones with xyz = 0 and the non-trivial one (x, y, z) = (± 3, −2, 1).
10 Citations

. We prove if A x + B y = C z , where A , B , C , x , y and z are positive integers, x , y and z are all greater than 2, then A , B and C must have a common prime factor. In this way, we demonstrate

### The generalized Fermat equation with exponents 2, 3, $n$

• Mathematics
Compositio Mathematica
• 2019
We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$ , to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the

### On ternary Diophantine equations of signature(p,p,2)over number fields

• Mathematics
TURKISH JOURNAL OF MATHEMATICS
• 2020
Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent

### A Simple and General Proof of Beal’s Conjecture (I)

Using the same method that we used in [1] to prove Fermat’s Last Theorem in a simpler and truly marvellous way, we demonstrate that Beal’s Conjecture yields—in the simplest imaginable manner, to our

### The Complexity of Mathematics

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be

### The Complexity of Mathematics

• Mathematics, Philosophy
• 2020
The strong Goldbach’s conjecture states that every even integer greater than 2 can be written as the sum of two primes. The conjecture that all odd numbers greater than 7 are the sum of three odd

### The Complexity of Number Theory

It is shown that if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Riemann hypothesis is true.

### The Complexity of Number Theory

• Mathematics
• 2020
The Goldbach’s conjecture has been described as the most diﬃcult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two

### On Upper Bounds with $ABC=2^m p^n$ and $ABC= 2^m p^n q^r$ with $p$ and $q$ as Mersenne or Fermat Primes.

Per the $ABC$ conjecture, for every $\varepsilon > 0$, there exist finitely many triples $\{A, B, C\}$, satisfying $A + B = C, gcd(A,B) = 1, B > A \geq 1$, such that $C > rad(ABC)^{1 + \varepsilon}$,

### Proof of the Tijdeman-Zagier Conjecture via Slope Irrationality and Term Coprimality

• Mathematics
• 2021
The Tijdeman-Zagier conjecture states no integer solution exists for AX + BY = CZ with positive integer bases and integer exponents greater than 2 unless gcd(A,B,C) > 1. Any set of values that

## References

SHOWING 1-10 OF 23 REFERENCES

### Partial descent on hyperelliptic curves and the generalized Fermat equation x3+y4+z5=0

• Mathematics
• 2011
Let C: y2=f(x) be a hyperelliptic curve defined over ℚ. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f1 f2 … fr. We shall define a ‘Selmer set’

### Primitive Integral Solutions to x 2 + y 3 = z 10

We classify primitive integer solutions to x^2 + y^3 = z^10. The technique is to combine modular methods at the prime 5, number field enumeration techniques in place of modular methods at the prime

### Twists of X(7) and primitive solutions to x^2+y^3=z^7

• Mathematics
• 2005
We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on

### Explicit descent for Jacobians of cyclic coevers of the projective line.

• Mathematics
• 1997
We develop a general method for bounding Mordell-Weil ranks of Jacobians of arbitrary curves of the form y p = f(x). As an example, we compute the Mordell-Weil ranks over Q and Q( p 3) for a

### Implementing 2-descent for Jacobians of hyperelliptic curves

• M. Stoll
• Mathematics, Computer Science
• 2001
This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2-Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus

### Explicit Chabauty over number fields

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K of degree d, and denote its Jacobian by J . Denote the Mordell–Weil rank of J(K) by r. We give an

### Finding rational points on bielliptic genus 2 curves

• Mathematics
• 1999
Abstract:We discuss a technique for trying to find all rational points on curves of the form Y2=f3X6+f2X4+f1X2+f0, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2.

### The Mordell-Weil sieve : proving non-existence of rational points on curves

• Mathematics
• 2010
We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how

### Descent on superelliptic curves

This paper defines a set which encapsulates information about local solubility of a particular collection of covers of a curve defined by an equation of the form y^q=f(x) and shows how to compute this set.