Finite Sum & Divisibility

I wish to discuss a small problem from The USSR Olympiad Problem Book (problem 59) about the finite sum of harmonic series. The problem asks us to prove that

 can never be an  integer for any value of .

I myself couldn’t think much about how to prove such a statement. So by reading the solution, I realised that how a simple observation about parity leads to this conclusion.

Firstly, observe that among the natural numbers from 2 to there is exactly one natural number which has the highest power of 2 as its divisor. Now, while summing up the reciprocals of these natural numbers we will get a fraction as the answer. In that fraction, the denominator will be an even number since it’s the least common multiple of all numbers from 2 to . And the numerator will be an odd number since it’s the sum of even numbers with one odd number (corresponding to the reciprocal of the number with the highest power of 2 as the factor). Since under no circumstances an even number can completely divide an odd number, denominator can’t be a factor of the numerator. Hence the fraction can’t be reduced to an integer and the sum can never be an integer.

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Solving Logarithmic Equations

While reading John Derbyshire’s Prime Obsession I came across the following statement (clearly explained on pp. 74):

Any positive power of eventually increases more slowly than any positive power of .

It is easy to prove this (existence) analytically, by taking derivative to compare slopes. But algebraically it implies that (for example):

There are either no real solution or two real solutions of the equation

for any given .

Now the question that arises is “How to find this ?” I had no idea about how to solve such logarithmic equations, so I took help of Google and discovered this Mathematic.SE post. So, we put and re-write the equation as:

Now to be able to use Lambert W function (also called the product logarithm function) we need to re-write the above equation, but I have failed to do so. 

But using WolframAlpha I was able to solve to get (which is an imaginary number, i.e. no real solution of this equation) but I was not able to figure out the steps involved. So if you have any idea about the general method or the special case of higher exponents, please let me know.

Cross Diagonal Cover – II

I will continue the previous post. Now the question to  be answered is that:

What percentage (or how many) of given grid can be covered by following Cross Diagonal Cover Algorithm?

I did some calculations, for example:

 

As per my calculations, I have following conjecture:

The following 3 cases are possible:

1. If , then squares will be covered.
2. If and both are odd, then squares will be covered.
3. If at least one of is even then squares will be covered.

As you can see that Case – 1 is quite trivial. I tried to prove other two cases for one week, but failed, so decided to post it as conjecture.

Nested Radical Sequences?

I ended up with a task at the end of my post last month: Resting with ‘Nested Radicals’

I have figured out that is a diverging series. But now I am struck with another problem:

What is the corresponding sequence for nested radical series?

In general, I am given a sequence and I write the corresponding series and check its convergence/divergence. But what if I am given a series without a general term (though wrong in many cases, as a given pattern of finite terms may be written as many different general terms.). For nested radicals I believe that there is only one general term and without this general term I will not be able to write terms of corresponding sequence (as in case of nested radicals I simply get partial sums). So I have a bigger question to ask:

Does there exists a sequence corresponding to any given series?