The Goldbach Conjecture and Lucky Numbers

I've written before about Lucky Numbers (Generating Lucky Numbers in Python and Lucky Numbers) as well as the Goldbach Conjecture (Goldbach's Conjecture and Zeckendorf's Theorem and Goldbach's Conjecture Revisited) but now it's time to combine the two topics. The reason we can do this is that primes and lucky numbers have similar distributions. The table in Figure 1 attests to this:

Figure 1: source URL

As stated on the site from which the table was taken:

What's most interesting about lucky numbers is the fact that they share a lot of properties with primes. As can be seen from the next table the density of the lucky numbers is close to the density of the primes. This seems also be true for the density of the twin luckies and the twin primes. In addition a lot of conjectures about primes seem also to be true for the luckies. For example one of the most famous ones, the Goldbach conjecture, stating that each even integer is the sum of at most two primes seems also to be true.

Goldbach decompositions are numerous and so, as with the decomposition into primes, we are interested in the minimal decomposition but first let's state the Goldbach conjecture for lucky numbers:

Every even number can be expressed as a sum of two lucky numbers

The smallest even number is 2 and that can expressed as 1 + 1. This is a little different to the primes where 1 is not regarded as a prime. Thus the Goldbach Conjecture for primes requires the even number to be greater than 2. The next even number is 4 and that can be expressed as 1 + 3 and so on. Let's take a number like 25800 and find it's minimal decomposition using SageMath. Figure 2 depicts the results using a screenshot from SageMathCell (permalink).

Figure 2: permalink

An interesting observation made on the website is that "no lucky number can have a digital root of 2, 5 and 8. This fact can sometimes be used to determine quickly that a given number is not lucky." I was able to find an explanation of why this is so thanks to a reference in Gardner's Workout by Martin Gardner. This is shown in Figure 3.

Figure 3

To see this, consider \(\frac{3k+2}{9}\). If \(k=1\) then the remainder is 5, if \(k=2\) then the remainder is 8, if \(k=3\) then the remainder is 2 and so on. The only remainders that can occur are 2, 5 and 8.

1 2 3 4 5 6 7 8 9
1 x 3 x 5 x 7 x 9 : first step of sieving process, all multiples of 2 are removed
1 x 3 x x x 7 x x : second step of sieving process, all multiples of 5 are removed

It is this sieving process, similar to the Sieve of Eratosthenes that causes the lucky numbers to have properties similar to the primes.

There are lots more "extensions" to the original Goldbach conjecture. One such one is the ternary Golbach conjecture described in the following abstract of a 79 page paper presented in 2014:

THE TERNARY GOLDBACH CONJECTURE IS TRUE

H. A. HELFGOTT 

Abstract. The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer n greater than 5 is the sum of three primes. The present paper proves this conjecture. 

Both the ternary Goldbach conjecture and the binary, or strong, Goldbach conjecture had their origin in an exchange of letters between Euler and Goldbach in 1742. We will follow an approach based on the circle method, the large sieve and exponential sums. Some ideas coming from Hardy, Littlewood and Vinogradov are reinterpreted from a modern perspective. While all work here has to be explicit, the focus is on qualitative gains. 

The improved estimates on exponential sums are proven in the author’s papers on major and minor arcs for Goldbach’s problem. One of the highlights of the present paper is an optimized large sieve for primes. Its ideas get reapplied to the circle method to give an improved estimate for the minor-arc integral.

A certain Zoltan Galantai has also investigated generalisations to the Goldbach conjecture at this site. 

The Golden Key

Figure 1: this is the book that mentions the
Golden Key, a description of its contents is
included at the end of this post

The sets of infinite natural numbers and infinite prime numbers are related by a formula given by Euler, which, famously known as the Golden Key, is given by:$$

\prod_{ p} \frac{1}{1-\displaystyle \frac{1}{p^{\,s}}}=\sum_n \frac{1}{n^{\, s}} \text{ where } s>1

$$where the left-hand-side products are carried over all the prime numbers \(p\) and the right-hand-side sum is carried over all the natural numbers \(n\).

For the range of numbers from 1 to 100,000 (with \(k\)=100,000), the results are as follows:$$
\prod_{p=2}^k \frac{1}{1-\displaystyle \frac{1}{p^{\, s}}}=1.64493274720203 \text{ and } \sum_{n=1} ^{k} \frac{1}{n^s}=1.64492406679823

$$The right hand side of Euler's formula is of course the Riemann Zeta function and so the equation can be rewritten as:$$

\zeta(s)=\sum_{n \geq 1}n^{-s}=\prod_p (1-p^{-s})^{-1}

$$

Figure 2

which is an easier form to remember (s can be any complex number with ℜs>1). The relationship at first seems strange, linking as it does a sum involving the reciprocals of the natural numbers and a product involving the reciprocals of the prime numbers. However, as this blog post points out, the formula is nothing but a fancy way of writing out the Sieve of Eratosthenes. The post goes on to derive the formula. I've just taken a screenshot of the working (Figure 2) rather than type it all out using LaTeX (lazy I know). In fact, this post is only the first in a long series of posts (from September 2013 to May 2017) dealing with Understanding the Riemann Hypothesis.

I came across the Golden Key when perusing Kumar Asok Mallik's book The Story of Numbers during his introduction to prime numbers on page 23. This is quite an interesting book that I've added to my Calibre library. The description of the book in the metadata is as follows:

This book is more than a mathematics textbook. It discusses various kinds of numbers and curious interconnections between them. Without getting into hardcore and difficult mathematical technicalities, the book lucidly introduces all kinds of numbers that mathematicians have created. Interesting anecdotes involving great mathematicians and their marvellous creations are included. The reader will get a glimpse of the thought process behind the invention of new mathematics. 

Starting from natural numbers, the book discusses integers, real numbers, imaginary and complex numbers and some special numbers like quaternions, dual numbers and p-adic numbers. Real numbers include rational, irrational and transcendental numbers. Iterations on real numbers are shown to throw up some unexpected behaviour, which has given rise to the new science of "Chaos". Special numbers like e, pi, golden ratio, Euler's constant, Gauss's constant, amongst others, are discussed in great detail.The origin of imaginary numbers and the use of complex numbers constitute the next topic. 

It is shown why modern mathematics cannot even be imagined without imaginary numbers. Iterations on complex numbers are shown to generate a new mathematical object called 'Fractal', which is ubiquitous in nature. Finally, some very special numbers, not mentioned in the usual textbooks, and their applications, are introduced at an elementary level.The level of mathematics discussed in this book is easily accessible to young adults interested in mathematics, high school students, and adults having some interest in basic mathematics. The book concentrates more on the story than on rigorous mathematics.

If I can read an entry a day from this book, I'll soon be a wiser man mathematically. Here is a link to a very useful series of slides explaining the importance of the Riemann zeta function and also mentioning the Golden Key.

on January 16th 2021

mainly improving the look of the mathematical expressions