Related papers: Denesting cubic radicals

Theorem. An irreducible cubic polynomial with rational coefficients has a root in a one step radical extension of Q if and only if the discriminate is a square of a rational number. Theorem. An irreducible polynomial x^4+px^2+qx+s with…

Let $p$ be an odd natural number $\ge 3$. Inspired by results from Euclid's {\em Elements}, we express the irrational $$y=\sqrt[p]{d+\sqrt R}, $$ whose degree is $2p$, as a polynomial function of irrationals of degrees $\le p$. In certain…

For $A\subseteq \{1, 2, \ldots\}$, we consider $R(A)=\{a/b: a, b\in A\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values assumed by a cubic form. We study this problem…

Let $p$ be a prime greater than $3$ and let $a$ be a rational p-adic integer. In this paper we try to determine $\sum_{k=1}^{[p/3]}\binom{3k}ka^k\pmod p$, and real the connection between cubic congruences and the sum…

We show that all perfect odd integer squares not divisible by 3, can be usefully written as sqrt(N) = a + 18p, where the constant a is determined by the basic properties of N. The equation can be solved deterministically by an efficient…

In this paper we show that Cardanos formula for the solution of cubic equations can be reduced to expressions involving only square roots if the real root is rational.

For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…

Given a real quadratic integer $u=A+B\sqrt{D}$ with cubic norm, we identify all the classes in a related form class group that represent primes $p$ for which $u$ is a cubic residue mod $p$. A special case of this result was conjectured in a…

In this paper, the Hermite problem has been approached finding a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In other words, the problem of writing cubic irrationals as a…

Let $a,b,c$ be distinct positive integers such that $a<b<c$ and $\gcd(a,b,c)=1$. For any non-negative integer $n$, the denumerant function $d(n;a,b,c)$ denotes the number of solutions of the equation $ax_1+bx_2+cx_3=n$ in non-negative…

In this paper, a new criterion is given to determine the $p-$rationality of some complex cubic number fields in terms of $ p-$divisibility of certain terms of a third-order recurrence sequence, several illustrated examples are…

The quintic equation with real coefficients $$x^5+5ax^3+5a^2x+b=0$$ is solved in terms of radicals and the results used to sum a hypergeometric series for several arguments.

We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3.…

In this paper we present some results related to the problem of finding periodic representations for algebraic numbers. In particular, we analyze the problem for cubic irrationalities. We show an interesting relationship between the…

We investigate the following problem: what is the smallest possible distance between a cubic irrational $\xi$ and a rational number $p/q$ in terms of the height $H(\xi)$ and $q$? More precisely, we consider the set $D_{3,1}$ consisting of…

We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be…

In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1…

Let $p$ be a prime congruent to 1 or 3 modulo 8 so that the equation $p=a^2+2b^2$ is solvable in integers. In this paper, we obtain closed-form expressions for $a$ and $b$ in terms of Jacobsthal sums. This is analogous to a classical…

In this survey, we studied the possibility of finding rational solutions to the equation $Ax^p+By^p+Cz^p=0$ via its attached hyperelliptic curve $Y^2=X^p+A^2(BC)^{p-1}/4$ and its rational points computed using computational tools.

A systematic study of the trigonometric equation A tan a + B sin b = C, where A, B and C^2 are rational numbers. The special case tan Pi/11 + 4 sin 3 Pi/11 = sqrt[11] appears in the classical literature.