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In this paper we determine the perfect powers that are sums of three fifth powers in an arithmetic progression. More precisely, we completely solve the Diophantine equation $$ (x-d)^5 + x^5 + (x + d)^5 = z^n,~ngeq 2, $$ where $d,x,z in mathbb{Z}$ and $d = 2^a5^b$ with $a,bgeq 0$.
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In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation $(x+1)^3 - (x+2)^3 + cdots - (x + 2d)^3 + (x + 2d + 1)^3 = z^p$, where $p$ is prime and $x,d,z$ are integers with $1 leq d leq 50$.
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 le r le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The so-called binomial sums are also considered. The problem of constructing polynomials that allow to calculate the values of the corresponding sums in certain cases is solved.
In the past two decades, many researchers have studied {it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${mathbb Z}/m{mathbb Z}$ for the order $m$ of the multiplicative character involved. A complete solution to the problem of evaluating index $2$ Gauss sums was given by Yang and Xia~(2010). In particular, it is known that some nonzero integral powers of the Gauss sums in this case are in quadratic fields. On the other hand, Chowla~(1962), McEliece~(1974), Evans~(1977, 1981) and Aoki~(1997, 2004, 2012) studied {it pure} Gauss sums, some nonzero integral powers of which are in the field of rational numbers. In this paper, we study Gauss sums, some integral powers of which are in quadratic fields. This class of Gauss sums is a generalization of index $2$ Gauss sums and an extension of pure Gauss sums to quadratic fields.
Let $ A subset B$ be rings. An ideal $ J subset B$ is called power stable in $A$ if $ J^n cap A = (Jcap A)^n$ for all $ ngeq 1$. Further, $J$ is called ultimately power stable in $A$ if $ J^n cap A = (Jcap A)^n$ for all $n$ large i.e., $ n gg 0$. In this note, our focus is to study these concepts for pair of rings $ R subset R[X]$ where $R$ is an integral domain. Some of the results we prove are: A maximal ideal $textbf{m}$ in $R[X]$ is power stable in $R$ if and only if $ wp^t $ is $ wp-$primary for all $ t geq 1$ for the prime ideal $wp = textbf{m}cap R$. We use this to prove that for a Hilbert domain $R$, any radical ideal in $R[X]$ which is a finite intersection of G-ideals is power stable in $R$. Further, we prove that if $R$ is a Noetherian integral domain of dimension 1 then any radical ideal in $R[X] $ is power stable in $R$, and if every ideal in $R[X]$ is power stable in $R$ then $R$ is a field. We also show that if $ A subset B$ are Noetherian rings, and $ I $ is an ideal in $B$ which is ultimately power stable in $A$, then if $ I cap A = J$ is a radical ideal generated by a regular $A$-sequence, it is power stable. Finally, we give a relationship in power stability and ultimate power stability using the concept of reduction of an ideal (Theorem 3.22).
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