I already have a few papers that show how Lill’s method can be used to solve quadratic equations. In my paper “Lill’s Method and Graphical Solutions to Quadratic Equations” I showed 2 methods that can solve quadratics with real roots and 1 method for solving quadratics with complex roots. That paper also mentions that the… Continue reading Quadratics with Complex Roots, Lill’s Circle and the Hyperbola
A Note on Whittaker’s Root Series Formula
If you want to learn about Whittaker’s Root Series Formula see my post “Whittaker’s Root Series: Going Transcendental”. That post has useful links to introductory material on the topic. You can also see my OEIS sequences, since a lot of them were obtained using Whittaker’s formula. Whittaker’s Root Series Formula is obtained using the coefficients… Continue reading A Note on Whittaker’s Root Series Formula
Littlewood Polynomials of Degree n with Closed Lill Paths
In this post I attempt to answer this question: How many Littlewood polynomials of degree n have a closed Lill path? A Littlewood polynomial is a polynomial that has all its coefficients equal to 1 or -1. Littlewood polynomials seem to be studied for their properties related to autocorrelation. A polynomial has a closed Lill… Continue reading Littlewood Polynomials of Degree n with Closed Lill Paths
The Tridecagon, Hyperbola and Lill’s Method
The regular tridecagon is another regular polygon that cannot be constructed using a compass and straightedge. In this post I want to show how the tridecagon can be constructed using the intersection of a circle and a hyperbola. In my previous posts “The Heptagon, Hyperbola and Lill’s Circle” and “The Nonagon, Hyperbola and Lill’s Method”… Continue reading The Tridecagon, Hyperbola and Lill’s Method
Adding and Subtracting the First n Prime Numbers to Get the Next Prime Number
In this post I want to present the following conjecture related to odd-indexed prime numbers (see OEIS sequence A031368): All the odd-indexed prime numbers larger than 2 can be obtained from all the previous prime numbers using addition and subtraction. The conjecture was inspired by Curiosa 67 from Scripta Mathematica ( page 159, Volume 7,… Continue reading Adding and Subtracting the First n Prime Numbers to Get the Next Prime Number
Lill’s Method, Prime Numbers and Tangent of Sum of Angles
In this post I want to explore again the property discussed in my paper “Lill’s Method and the Sum of Arctangents”. I’ll apply the property to this question: If tan(θ1)=2, tan(θ2)=3,tan(θ3)=5,…,tan(θn)=n-th prime number, then what is tan(θ1 + θ2 +… θn)? The question can be easily solved with a calculator. We’ll see that the answer… Continue reading Lill’s Method, Prime Numbers and Tangent of Sum of Angles
Fregier Quarter Point and the Focus of the Parabola
If you are not familiar with Frégier’s Theorem then you should read my introductory post on the topic “Frégier’s Theorem and Frégier Points”. Later, I also wrote a post about an alternative way of finding the Fregier points for a parabola. The property or theorem discussed in this post is only relevant to parabolas. I… Continue reading Fregier Quarter Point and the Focus of the Parabola
The Nonagon, Hyperbola and Lill’s Method
The nonagon is another polygon that cannot be constructed with ruler and the compass (see OEIS sequence A004169). However, the nonagon can be constructed using conics (see OEIS sequence A051913). In this post I want to show how we can use the intersection of the Lill circle of the polynomial x3 – 0.75x + 0.125… Continue reading The Nonagon, Hyperbola and Lill’s Method
Adding and Subtracting 1 and the First n Primes to Get the Next Prime (Part 2)
In part 1 I dealt with even-indexed primes. In this post I want to show the first few examples of how to obtain the odd-indexed prime numbers using the addition or subtraction of 1 and all the other preceding primes. The odd-indexed primes are covered by the OEIS sequence A031368. The summation for each prime… Continue reading Adding and Subtracting 1 and the First n Primes to Get the Next Prime (Part 2)
Adding and Subtracting 1 and the First n Primes to Get the Next Prime (Part 1)
In this post I want to show a few examples of how a prime number can be obtained from all the preceding prime numbers and 1 using addition and subtraction. In Part 1 I’ll deal with only even-indexed primes: a(n)=prime(2n). 3 is the first even-indexed prime, 7 is the second, 13 is the third, 19… Continue reading Adding and Subtracting 1 and the First n Primes to Get the Next Prime (Part 1)