## 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)

## 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