The Prime Number Theorem (up to a constant factor)

It is suprisingly easy and pretty to give an elementary proof that the prime number theorem (that the number of primes less than is asymptotic to ) is true up to a constant factor. (I will address only the part where we show that there are many (i.e. at least ) primes less than .)

Consider a binomial coefficient . Let be a prime. We first want to know what the largest power of is which divides . There is more than one way to express the answer, the most elegant one I know of is as follows:

Add the numbers and in base (by the usual primary school algorithm). Then the number of carries that occur in this addition is equal to the greatest power of dividing .

What is most important for us is the fact that this number is no more than .

We immediately get

which is of course equivalent to

Make a good choice of and we are within a constant factor of the prime number theorem!

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Concyclicity in the Euclidean Plane

I want to address the question of how to determine whether or not four points are concyclic. In a previous post, this question reared its head and was solved by an introduction of the circumcentre of three of the points. In this post, I present a direct approach.

We prove the following:

Let be four points in . Then these four points are concyclic if and only if

For the proof, first suppose that the points all lie on the curve

The four equations we obtain give a linear dependence of the columns in the matrix under consideration, hence its determinant is zero.

Conversely, suppose that the determinant in question is zero. Then there is a nonzero vector in the nullspace of our matrix. If its first entry is nonzero, we can normalise it to be of the form and we obtain the circle that the four points lie on. If the first entry of this vector is equal to zero, then the four points must lie on a line, which after is all is just a degenerate circle, QED.