Prime Numbers 1!
We all hear about these numbers, scum bags to be honest so arrogant that they don’t like to be shared, what I mean by share is that these numbers can’t be divided by any other number other than themselves. And of course the unity or 1. What made me curious about these numbers is the fact that when you look at their sequence like the one mentioned below it’s hard to find something that our incredible brain is so good at finding.
Yes, I am talking about patterns. We as humans have this incredible sense of finding, sequence or patterns. We’ll talk about that some other day but for now, just know that we can find patterns. But I want to ask you can you find any pattern in that given set of prime numbers?
Well, some of you might and some of you might not. But for the better perspective of the few people who are reading this post of mine, I’ll help you guys out. Let’s take the first 10 prime numbers over here. Now for proving every part I’ll try to use Python-Programming. In order to compute. There are various conjectures, now to understand these I want you to know what is actually a conjecture.
Conjecture:- Conjecture is nothing but a conclusion that is subjected to be true because of the given evidence but, there isn’t enough proof available that would be enough to prove it for a theorem.
For example:- In Indian society especially in Hindus, fasting of karvachauth is done by the wife for her husband’s long life, well the idea seems good and is considered to be true hence so many people participate in this ritual, however, because of the inadequate data provided on this particular topic, it would be pragmatically inaccurate to consider this assumption as true. Or what if she marries someone else, would she be helping out two husbands? Now, these analogies might’ve cleared out your doubts on the matter of conjecture as it is really necessary to understand this in order to consume what’s about to come.
So similar to the example above prime numbers have their own rituals, these rituals for the given time are considered to be true but God knows if they’re true or false. Given the fact that for a conjecture to be considered a theorem it should be absolute. Like the universal truth of life and death.
Gilbreath’s conjecture:- This one is really close to me, see i discovered this one myself, not discovered of course I came across this beautiful conjecture that goes un notice unless you’re trying to fool around with prime, unlike the Goldbach’s conjecture, which we’ll discuss further on it’s not that well known. So here’s a pattern for the conjecture now pause for a while and try to understand what is did…
2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29……
1 , 2 , 2 , 4 , 2 , 4 , 2 , 4, 6 ……
1 , 0 , 2 , 2 ,2 ,2………
So the conjectures say like if you take n number of prime numbers, that means from 2 to any positive integer which satisfies being a prime number is considered. Now you subtract the next number in the series from the previous number and after that, you take its modulus meaning, even if the value is negative you need to take it as positive, for example…
| 20–90 | = 70 which is considered as positive.
And now with the available series, you do the same what you’ll reach is that difference when iterated again and again no matter how long the series was would lead to always 1. No matter how you do it, if you follow the rules it will always lead to 1. Now how is this astonishing some might ask, and the fact is that yes i think you’re absolutely right to question this, but just think about this,
Prime numbers don’t have a pattern, they’re rude they’re harsh, they could lie just after one another like the 2 and 3 or they could be far apart from each other like 29 and 23 their occurrence is random, but then too they tend to bend down to the basics of mathematics like an ice cube in water, simply, subtly but creating a huge impact at the same time. The conjecture not only questions the randomness of these prime numbers but also questions a human’s ability to see a bigger picture. To understand this furthermore, I did something new, it made me realize the weakness in python as a language and more importantly introduced me to golang, but to keep things simple let’s go with python3.
I wanted to check if the conjecture was true for all prime numbers from 2 to 200000. Yes, yes it holds for the 200000. But the pattern in gaps gets more random as we start to move towards 200000, the gaps between the numbers increases up to 80, which means no prime number in the next 80 digits, the most frequent difference is 2.
One interesting pattern is that somehow if you don’t take the modulus and try to find the value of the pattern, you can easily find it using the pascals triangle.
There were many notable attempts to prove this conjecture but every single one of them went in vain, but this beautiful problem, not well known I think holds all the mysteries of prime as it questions the randomness of primes and their locations on the number line. This article is a series so the next one will be continued from here on. In the next one, I’ll be writing about the code for these conjectures and patterns formed because of them.