Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the equation $x^2+dy^6=z^p$ for square-free $1le dle 20$
87
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The purpose of the present article is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation $x^2+dy^6=z^p$ for square-free values $1 le d le 20$ following the approach of [PT]. The main innovation is to make use of the symplectic argument over ramified extensions to discard solutions, together with a multi-Frey approach to deduce large image of Galois representations.
rate research
Read More
Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z}, ngeq 2, xyz eq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Bakers theory and Bilu-Hanrot-Voutiers result on primitive divisors of Lucas numbers.
In this paper, we prove that the only primitive solutions of the equation $a^2+3b^6=c^n$ for $ngeq 3$ are $(a,b,c,n)=(pm 47,pm 2,pm 7,4)$. Our proof is based on the modularity of Galois representations of $mathbb Q$-curves and the work of Ellenberg for big values of $n$ and a variety of techniques for small $n$.
In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $ngeq 2$ and $d=p^b$, $p$ a prime and $pleq 10^4$. The main ingredients are the characterization of primitive divisors on Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)inmathbb Z[x]$, $a$ a positive integer and $b$ an arbitrary positive integer.
Let $E$ be an elliptic curve over $Q$. It is well known that the ring of endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary reduction, is an order of the quadratic imaginary field $Q(pi_p)$ generated by the Frobenius element $pi_p$. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of $a_p^2-4p$, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function $pi_{E,r,h}(x)$ counting the number of primes $p$ up to $x$ such that $a_p^2-4p$ is square-free and in the congruence class $r$ modulo $h$. We give in this paper the precise asymptotic for $pi_{E,r,h}(x)$ when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the Lang-Trotter conjecture, and with some other problems related to the curve modulo $p$.
We solve the diophantine equations x^4 + d y^2 = z^p for d=2 and d=3 and any prime p>349 and p>131 respectively. The method consists in generalizing the ideas applied by Frey, Ribet and Wiles in the solution of Fermats Last Theorem, and by Ellenberg in the solution of the equation x^4 + y^2 = z^p, and we use Q-curves, modular forms and inner twists. In principle our method can be applied to solve this type of equations for other values of d.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
