اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the largest prime factor of $n^2+1$
129
0
0.0
(
0
)
تأليف
Jori Merikoski
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harmans sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selbergs eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
قيم البحث
اقرأ أيضاً
Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)cdots f(m)$. We prove that if $m > max{10^{12},4^{n+1}}$, then there exists a prime divisor $p$ of $P_{m,n
}$ such that ord$_{p}(P_{m,n} )leq ncdot 2^{n-1}$. For $n=2$, we establish that for every positive integer $m$, there exists a prime divisor $p$ of $P_{m,2}$ such that ord$_{p} (P_{m,2}) leq 4$. Consequently, $P_{m,2}$ is never a fifth or higher power. This extends work of Cilleruelo who studied the case $n=1$.
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 p
roofs are Bakers theory and Bilu-Hanrot-Voutiers result on primitive divisors of Lucas numbers.
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.
In this paper, we determine all the squares in the sequence ${prod_{k=2}^n(k^2-1)}_{n=2}^infty $. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the $p$-adic valuation of the terms in this sequence.
We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 geq p^theta$ for $theta=1/2+1/2000.$ This improves the work of Matomaki (2009) who obtained the result for $theta=1/2-varepsilon$ (with the added constrai
nt that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $theta=4/9-varepsilon.$ Similarly as in the work of Matomaki, we apply Harmans sieve method to detect primes $p equiv 1 , (d^2)$. To break the $theta=1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument as Zhang (2014) used to obtain equidistribution beyond the Bombieri-Vinogradov range for primes with respect to smooth moduli. To optimize the argument we incorporate technical refinements from the Polymath project (2014). Since the moduli are squares, the method produces complete exponential sums modulo squares of primes which are estimated using the results of Cochrane and Zheng (2000).
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
