Variations on Ramanujan's nested radicals (original) (raw)
On an entry of Ramanujan in his Notebooks: a nested roots expansion
Journal of Computational and Applied Mathematics, 2005
In this letter, the elementary result of Ramanujan for nested roots, also called continued or inÿnite radicals, for a given integer N , expressed by him as a simple sum of three parts (N = x + n + a) is shown to give rise to two distinguishably di erent expansion formulas. One of these is due to Ramanujan and surprisingly, it is this other formula, not given by Ramanujan, which is more rapidly convergent!
On a class of Labesgue-Ramanujan-Nagell equations
arXiv: Number Theory, 2020
We investigate the Diophantine equation cx2+d2m+1=2yncx^2+d^{2m+1}=2y^ncx2+d2m+1=2yn in integers x,ygeq1,mgeq0x, y\geq 1, m\geq 0x,ygeq1,mgeq0 and ngeq3n\geq 3ngeq3, where ccc and ddd are given coprime positive integers such that cdnotequiv3pmod4cd\not\equiv 3 \pmod 4cdnotequiv3pmod4. We first solve this equation for prime nnn, under the condition nnmidh(−cd)n\nmid h(-cd)nnmidh(−cd), where h(−cd)h(-cd)h(−cd) denotes the class number of the quadratic field mathbbQ(sqrt−cd)\mathbb{Q}(\sqrt{-cd})mathbbQ(sqrt−cd). We then completely solve this equation for both ccc and ddd primes, under the assumption gcd(n,h(−cd))=1\gcd(n, h(-cd))=1gcd(n,h(−cd))=1. We also completely solve this equation for c=1c=1c=1 and dequiv1pmod4d\equiv1 \pmod 4dequiv1pmod4, under the condition gcd(n,h(−d))=1\gcd(n, h(-d))=1gcd(n,h(−d))=1. For some fixed values of ccc and ddd, we derive some results concerning the solvability of this equation.
Review of Radical solution of a general degree n polynomial equation
In this research the general polynomial equation is investigated for a solution. The solution method is based on a previous finding by the author of this paper that asserts that any number other than zero has infinite number of algebraic factorizations. This finding makes it possible for an algebraic extension to be radical tower over a field of algebraic numbers. This means polynomial equations in general have solvable Galois groups. Thus equations of degree five and above have general radical formulae for their solution. Keywords: Radical solution of the general polynomial equation; an infinite number of algebraic factorization of a number.
A Continued Fraction of Ramanujan and Some Ramanujan-Weber Class Invariants
2017
On Page 36 of his ``lost" notebook, Ramanujan recorded four qqq-series representations of the famous Rogers-Ramanujan continued fraction. In this paper, we establish two qqq-series representations of Ramanujan's continued fraction found in his ``lost" notebook. We also establish three equivalent integral representations and modular equations for a special case of this continued fraction. Furthermore, we derive continued-fraction representations for the Ramanujan-Weber class invariants gng_ngn and GnG_nGn and establish formulas connecting gng_ngn and GnG_nGn. We obtain relations between our continued fraction with the Ramanujan-G\"{o}llnitz-Gordon and Ramanujan's cubic continued fractions. Finally, we find some algebraic numbers and transcendental numbers associated with a certain continued fraction A(q)A(q)A(q) which is related to Ramanujan's continued fraction F(a,b,lambda;q),F(a,b,\lambda;q),F(a,b,lambda;q), the Ramanujan-G\"{o}llnitz-Gordon continued fraction H(q)H(q)H(q) and the Dedekind eta fun...
A Note on Kurosh Amitsur Radical and Hoehnke Radical
Thai Journal of Mathematics, 2011
Abstract: The notion of Radical classes is introduced in [1]. We prove here some useful equivalent conditions for a subclass of a fixed universal class to be a radical class. We introduce the notion of Hoehnke radical and give some consequences of Hoehnke radical ...
Bulletin of the Australian Mathematical Society, 2011
A radical ρ is called prime-like if for every prime ring A, the polynomial ring A[x] is ρ-semisimple. Let α be a radical satisfying the polynomial equation α(A[x])=(α(A))[x] for every ring A. A radical γ is called α-like if for every α-semisimple ring A, the polynomial ring A[x] is γ-semisimple. In this paper, we study properties of α-like radicals. We show that α-likeness is a generalization of prime-likeness and extend some results concerning prime-like radicals. This allows us easily to find distinct special radicals which coincide on simple rings and on polynomial rings, which answers a question put by Ferrero.
Some experiments with Ramanujan-Nagell type Diophantine equations
Glasnik Matematicki, 2014
Stiller proved that the Diophantine equation x 2 + 119 = 15 • 2 n has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type x 2 = Ak n + B with many solutions. Here, A, B ∈ Z (thus A, B are not necessarily positive) and k ∈ Z ≥2 are given integers. In particular, we prove that for each k there exists an infinite set S containing pairs of integers (A, B) such that for each (A, B) ∈ S we have gcd(A, B) is square-free and the Diophantine equation x 2 = Ak n + B has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form x 2 = Ak n + B with k > 2, each containing five solutions in non-negative integers. We also find new examples of equations x 2 = A2 n + B having six solutions in positive integers, e.g. the following Diophantine equations has exactly six solutions: