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

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)

A Prime Counting Sequence and Andrica’s Conjecture

In this post I want to discuss a prime counting sequence similar to OEIS sequence A066888. The sequence A066888 counts the number of primes between 2 consecutive triangular numbers. On the OEIS page of the sequence there is a conjecture that says that there is at least one prime number between 2 consecutive triangular numbers.… Continue reading A Prime Counting Sequence and Andrica’s Conjecture

The Heptagon, Hyperbola and Lill’s Circle

The heptagon cannot be constructed with just a ruler and a compass. However, in this post I’ll show how you can construct the heptagon using a hyperbola and the Lill’s circle of a third degree polynomial. The heptagon construction is very similar in nature to my trisection construction that I presented in my “Trisection Hyperbolas… Continue reading The Heptagon, Hyperbola and Lill’s Circle

Trisection Hyperbolas and Lill’s Circle

In my previous blog post about trisection “Angle Trisection: a Neusis Construction using Lill’s Method and Lill’s Circle”, I showed a “mechanical” method for trisecting an angle smaller than 90 degrees. The neusis construction involved the Lill’s method representation of cubic equations of the form x3  -3tan(θ)x2 -3x + tan(θ). These cubic equations have 3 real… Continue reading Trisection Hyperbolas and Lill’s Circle