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Register a new userOn the solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ for $d$ a prime power
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Authors
Angelos Koutsianas
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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.
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In this paper, we solve the equation of the title under the assumption that $gcd(x,d)=1$ and $ngeq 2$. This generalizes earlier work of the first author, Patel and Siksek [BPS16]. Our main tools include Frey-Hellegouarch curves and associated modular forms, and an assortment of Chabauty-type techniques for determining rational points on curves of small positive genus.
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.
Let $f(x)=x^{2}(x^{2}-1)(x^{2}-2)(x^{2}-3).$ We prove that the Diophantine equation $ f(x)=2f(y)$ has no solutions in positive integers $x$ and $y$, except $(x, y)=(1, 1)$.
Assume a polynomial-time algorithm for factoring integers, Conjecture~ref{conj}, $dgeq 3,$ and $q$ and $p$ are prime numbers, where $pleq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $log(q)$ that lifts every $mathbb{Z}/qmathbb{Z}$ point of $S^{d-2}subset S^{d}$ to a $mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $mathbb{Z}/qmathbb{Z}$ points of $S^{d-2}subset S^d$.
We consider the distribution of normalized Frobenius traces for two families of genus 3 hyperelliptic curves over Q that have large automorphism groups: y^2=x^8+c and y^2=x^7-cx with c in Q*. We give efficient algorithms to compute the trace of Frobenius for curves in these families at primes of good reduction. Using data generated by these algorithms, we obtain a heuristic description of the Sato-Tate groups that arise, both generically and for particular values of c. We then prove that these heuristic descriptions are correct by explicitly computing the Sato-Tate groups via the correspondence between Sato-Tate groups and Galois endomorphism types.
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