Consider the equation 
. You might notice that 
, but no other solutions occur immediately (except when 
, which we dismiss as trivial). We assume then (without loss of generality) that 
. The following is a short and elegant proof that 
, 
is the only solution. Read the rest of this entry »
Solving inverted power equation with analysis
July 20th, 2011Four frogs on a square
July 12th, 2011Four frogs are standing on a square, one on each vertex. You are allowed to move them as much as you want by making them jump. A frog can only jump over another frog, and it lands on that frog’s other side so that the distance between them is the same as before. Is it possible to make the frogs form a square of a different size than the original? Read the rest of this entry »
Euler’s product (part 4)
July 7th, 2011In this last part we will consider some applications of Euler’s product, such as infinitude of primes, sum of prime reciprocals and more. Read the rest of this entry »
Euler’s product (part 3)
July 3rd, 2011This third part is about the manipulation of the harmonic series into an expression that involves primes. As I have mentioned before, by doing this we essentially form a bridge that enables analysis to be applied to primes. That is in fact a desirable result by itself, for even without applications it is a nice display of linkage between seemingly distinct areas in mathematics. Read the rest of this entry »
Euler’s product (part 2)
May 17th, 2011We have previously noted the plausibility of a link between a sum involving the natural numbers and a product involving the primes. As it turns out it is not necessary to look very far for a sum, a simple general term would do. Though this may seem odd, it’s actually essential – the smallest change in the sum’s general term can lead to extraordinary complications in the product (if a suitable one exists). This second article is about the harmonic series. Read the rest of this entry »
