The Infinitude of Primes and the Sexiness of Sexy Primes

Once upon a time, a humble shepherd tended to his flock in the rolling hills of ancient Greece. One day, as he counted his goats, he realized that there was a certain number of goats that he just couldn't seem to divide into equal groups. He noticed that 20 goats could be divided into 5 groups of 4 each, or 2 groups of 10, or 4 groups of 5. But no matter how hard he tried, 17 goats could never be divided into groups. The numbers that could be divided were not as special, but the ones that couldn't—well, those were a mystery. They were like stubborn puzzle pieces that just wouldn't fit. That is how I like to imagine humans conceived the first rudimentary idea of primes.

Formally, a prime is a natural number greater than 1 that has only two positive divisors, one and itself. Prime numbers have always been at the center of attention for mathematicians. Their simple yet elusive nature has always led number theorists astray. What makes them special and worthy of attention?

Prime numbers, it turns out, are the building blocks of all other numbers. Just as atoms combine to form molecules and make up all matter, prime numbers can be combined through multiplication to form any positive integer greater than 1. This means that every positive integer greater than 1 can be expressed as a unique combination of prime numbers, making primes a fundamental concept in mathematics, especially in number theory, the study of the properties and relationships of numbers, particularly integers.

To better understand the distribution of primes along the number line, one would have to employ a tool called the sieve of Eratosthenes. The idea is simple: start with a list of all positive integers up to a certain limit, and then systematically eliminate all multiples of the primes, leaving only the primes. This process could then be repeated with larger and larger limits.

As seen in the diagram above, all the composite (non-prime) numbers have been crossed out leaving only primes.

Can it be possible that primes have a finite limit? Or do they continue on, unchanging and unbounded, for all eternity? The answer to this age-old question was given by Euclid of Alexandria in 300 BC. Euclid was a smart man who made significant contributions to geometry and arithmetic. One day, while meditating on the problem, Euclid had a brilliant insight. He realized that if there were only a finite number of primes, then he could form a new number by multiplying all of these primes together and adding 1. If this new number is not divisible by any of the primes, then it must be either a new prime number or it is divisible by a prime not included in our original finite list. This contradicts the assumption that there are only a finite number of primes, and thus proved that there must be an infinite number of primes.

Theorem: There are an infinite number of primes

Proof by contradiction

Assume there is only a finite list of primes P = { p1, p2, p1,...,pn}

Take a number N such that N = ( p1 * p2 * p1 *...*pn) + 1

Since N leaves a remainder of 1 when divided by any of the prime numbers P ={ p1, p2, p1,...,pn} , it is therefore not divisible by any of them. As we saw how primes are generated using the sieve of Eratosthenes, then the number N either factors itself making it a prime or it is factored by a prime not in the list.

Therefore we can always generate yet another prime number!

Common misconception: the number N is not necessarily a prime itself as some people may think. The proof merely points out that it can't be factored out by any of the primes in the product.

Takeaways

• Twin primes are special types of primes which differ by 2.(e.g. 3 and 5)

• Cousins primes are of the form ( P, P+ 4). In short they differ by 4.(e.g. 3 and 7)

• Sexy primes are of the form ( P, P+ 6). They differ by 6. (e.g. 5 and 11)

I don't know who came up with that term :)

Question: Can a twin prime be sexy( pun intended)?

Answer: yes, 5 differs from 3 by 2 and from 11 by 6. That makes it both a twin prime and a sexy prime.

The Twin prime conjecture states that there is an infinite number of twin primes. This conjecture still remains one of the unproven cornerstone problems in mathematics.

Yitang Zhang showed that there are infinite pairs of primes that have a gap of a fixed number less than or equal to 70,000,000 . Although 70 million is a large number, it still lowers the bounds greatly and shows that the gap between primes doesn't keep increasing indefinitely. Don't get it wrong, Yitang’s result does not state that there is at least one prime for every interval that is 70 million apart. It merely states that prime occurrences which differ by a fixed gap of 70 million or less are infinite in number.

Imagine a very long ruler that is 70 million units in length. If you repeatedly slide the ruler along the number line, it will eventually enclose at least two primes with a fixed gap an infinite number of times. However, it does not guarantee that there will always be two primes enclosed at any given moment.

It doesn't imply that “There is always a prime nearby.’’ instead it implies that “There can always be a prime nearby.’’

Mathematicians strive so hard to lower the bound in order to get a complete understanding of the distribution of primes along the number line.

Edit: James Maynard, a mathematician, published a paper recently which lowered the gap even further down to 600

• “Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.” - Leonhard Euler

• "The primes are the raw material out of which we build arithmetic." - G. H. Hardy

• If you want to explore more about the mysteries of the primes, you can watch this video of Professor Terrence Tao lecturing on “Structure and Randomness in the Prime Numbers.’’

***There's a number, prime and so true, it's indivisible, there's nothing it can't do, with factors just two, it's a marvel, that's true, infinite in might, no end to their sight***