I have always been thinking how mathematicians make discoveries. They come up with beautiful theorem and formulas that were unknown to mankind an hour ago until they discovered it first. Unlike the poets and writers, mathematicians cannot cook some stories or imagine some beautiful verses in the mind. So how does it work? When you are solving a problem you know where to begin, you know where to end and you will mostly have the tools that will help you solve the problem. Now let assume that you teacher enters to your class and instead of giving you a problem to solve from your course, asks you to discover something that you previously did not know. Now you don’t know where to begin where to end where to look for something new. So how do you go about making a mathematical discovery?
I am writing a series of article in this blog where I shall describe the thought process that I underwent as I made some interesting discoveries in analytical number theory. The purpose of this article is not to make a better problem solver but rather to encourage one to think like a mathematician who is on a journey of mathematical exploration and discovery. Therefore I shall not go into the precise technical details of the proofs of the results in this article but rather I shall give the sketch of the main ideas that led to these results and describe how one thought led to another and eventually culminated in a small theory comprising of many little discoveries along the way. Anyone interested in deriving the actual proof can easily reconstruct the proof from the central idea and thought process as well the descriptive details given for each result. The purpose of this article is to encourage the mathematical thought process that lead to new discoveries. In case there are any terminologies which I feel might be unfamiliar to my readers, I have provided hyperlink to a Wikipedia page on the same.
BEGINNING WITH A SIMPLE IDEA
SECTION 1. OBSERVATION
Many discoveries begin with an initial observation. For example thousands of years ago somebody might have observed that sum of the square of the sides of a right triangle equals the square of the hypotenuse. This observation stood the test of repeated experiments and only after that it would have inspired some ancient genius to establish this observation as a theorem by finding a rigorous proof. Similarly someone would have observed that the ratio of the circumference of a circle to its diameter remains constant and this eventually led to the discovery of . A keen observation power is therefore one of the prerequisites of making a mathematical discovery.
Many years ago when I first learned definite integrals as a school student, I developed an affinity for one particular theorem which allowed us to evaluate the limits of a sum using definite integrals. The formula was
Lemma 1. If is Riemann integrable in [0,1] then
Three years ago I came across a type of sequence called equidistributed sequences or uniformly distributed sequences. There I found the following result theorem that had striking resemblance to Lemma 1.
Lemma 2. If the sequence is uniformly distributed modulo 1 then
As soon as I saw this result on uniformly distributed sequences modulo 1, I knew that I had seen something like it before; the result in Lemma 1. Replacing in Lemma 1 by we get Lemma 2. The similarity between Lemma 1 and Lemma 2 should automatically make us wonder why they are so similar. So at this point what are the options that we have? Either it is a coincidence or there is some hidden mathematical relation that both these results are following. Since we are like explorers searching to make a discovery we would want to believe that this similarity is not a coincidence but because of some hidden mathematical relation that we seek to unearth.
SECTION 2. EXPLAINING THE OBSERVATIONS
Once we have found an observation to work on the next step is to find a mathematical explanation for the observation. If we can find an explanation within the existing mathematics then our observation is a new theorem or an extension or a generalization of one of the areas of existing mathematics. However if no known mathematics can give a satisfactory explanation of our observation then we will have to invent new mathematics all together.
Continuing with our example, what are the possible thoughts that can strike us when we try to explain the similarity between Lemma 1 and Lemma 2? The possible hypothesis are:
- Lemma 1 and Lemma 2 are two different representations of the same phenomenon.
- The sequence 1/n, 2/n, … , n/n approaches uniform distribution modulo 1 as .
To test our hypothesis we go back to the definition of uniform distribution modulo 1 and test if this definition applies to the sequence 1/n, 2/n, … , n/n as and we find that this actually is the case and our hypothesis is true.
SECTION 3. BUILD THE THEORY
So we have discovered that the sequence 1/n, 2/n, … , n/n approaches uniform distribution modulo 1 as . We shall use this starting point and a guiding star in our exploration. We now know that the theme of our exploration is going to be centered around sequences uniformly distributed modulo 1. The next logical questions that strikes (or rather should strike) our mind are:
- Yes the sequence uniform distribution modulo 1 but what about the sequence or in general ?
- This will lead us to formulate the question in the most general form, when is the sequence uniformly distributed modulo 1?
Again applying the definition of uniform distribution modulo 1, we can easily prove the following result. Let denote the greatest integer function. We have
Lemma 3. As the sequence , approaches uniform distribution modulo 1 if
for every t, .
SECTION 4. APPLY YOUR THEORY
So far so good but now we need show that our theory is valuable by showing its applications. We want to find the sequence that satisfy Lemma 3. It is easy to see that Lemma 3 implies for every . Hence grows much slower than any quadratic polynomial function of n. At this point we sit an recall all well know sequences that grow at this rate.
We can find many such sequences but one particular sequence that we cannot afford to miss out is the sequence of prime numbers. Recall that the prime number theorem implies that . We find that the sequence of primes satisfy Lemma 3. Similarly we can show that the sequence of composite numbers denoted by also satisfy Lemma 3.
We have found a lot of interesting insights so far. But can we be more adventurous? Notice that if two or more sequence satisfy Lemma 3 then their linear combination will also satisfy Lemma 3. So we can put together all our results so far in the form a beautiful and elegant theorem.
Theorem 1. Let be the n-th prime, be the n-th composite number and let and be any three constants such that then
Thus we see that starting with basic theorem on definite integrals and a fundamental theorem on uniform distribution modulo one, we by properly directing the thought process have ended up with a non-trivial theorem connecting prime numbers, composite numbers and natural numbers. Now we can let our imaginations run wild and we can use our theorem to derive many interesting results on prime or composites. For example
To read the second part of this post, click A mathematician’s mind-2.
NOTES AT THE END OF CHAPTER 1
In this chapter we saw how observing similarities between two different results of mathematics can reveal that both these results are special cases of a larger family of relations provides an ideal platform for beginning a mathematical exploration. In the second post on this topic in Chapter 2, we shall continue our exploration forward from here develop our theory further by extending our results to divergent series. The topic in this series of posts are based on my paper ‘On a unified theory of numbers‘ which I have co-authored with Prof. Marek Wolf, Institute of Theoritical Physics, Warsaw, Poland. Currently we are revising and re-formating our paper. If you spot any mathematical or typographical error please do let us know. Also any suggestion or comments to improve this blog are welcome.