In this post, I shall explore some of the aspects of prime numbers. For centuries, prime numbers have drawn attention to mathematicians, professionals and amateurs alike. Most stubborn attempts to find the complete picture of prime numbers proved fruitless.
A positive integer $p$ is a prime, if it has only two divisors, $1$ and $p$ itself. There is a very simple, but elegant proof of infiniteness of prime numbers, thanks to Euclid(which dates back to 4th century BCE). The proof is based on contradiction. If we assume that prime numbers are finite and last one of them is $q$, and define a number $n$ which leaves a remainder of $1$ when divided by all the primes $\le q$. Formally $n=2 \cdot 3 \cdot 5 \cdots q+1$. As $n$ is not divisible by any prime $\le q$, either $n$ itself is a prime, or there exists a prime $>q$ which divides $n$. Both contradicts our initial assumption about prime numbers being finite. Over time, many algorithms have been formulated to count primes, some of them are very complex and require deeper understanding of number theory. So I shall not discuss about them in this post.
Prime numbers of special form have been studied in great details. Most famous of these special primes are Mersenne primes and Fermat primes.
Mersenne primes are named after Father Marin Mersenne, a French priest. Mersenne numbers are defined as $M_n=2^n-1 \hspace{1 mm}, n\geq 1$. It is not yet known whether the set of Mersenne primes are finite or infinite.
Famous mathematician Pierre de Fermat believed that numbers of form $2^{2^n}+1$ are primes, based on first $5$ values ($0\le n \le 4$). But no other Fermat primes are not known for $n>4$.
A lot of conjectures are there concerning primes. Most notable of them are Goldbach conjecture and twin prime conjecture. First one states that every even integer greater than $2$ can be expressed as the sum of two primes.
A couple of prime numbers have even been associated with superstitions. Most famous of them is the number $13$, considered as unlucky. Fear of number $13$ even has a special name, Triskaidekaphobia (it was mentioned in my favorite TV series Friends - S08E20 :) ). Other superstitious prime is the Belphegor's prime - $1000000000000066600000000000001$. It has exactly $31$ digits, inverse of $13$. It contains $666$, 'the number of the beast' at the center and have $13$ zeros on either sides. And it is also a palindromic prime.
Primes also have practical applications. Randomness of prime numbers and difficulty of factorizing lie at the heart of several public encryption schemes, which are used to secure communication systems.
Primes will most probably befuddle us for many more years to come. Where is the fun if the everything is predictable? :) As English novelist Mark Haddon said, "Prime numbers is what is left when you have taken all the patterns away."
Thanks for reading :)
A positive integer $p$ is a prime, if it has only two divisors, $1$ and $p$ itself. There is a very simple, but elegant proof of infiniteness of prime numbers, thanks to Euclid(which dates back to 4th century BCE). The proof is based on contradiction. If we assume that prime numbers are finite and last one of them is $q$, and define a number $n$ which leaves a remainder of $1$ when divided by all the primes $\le q$. Formally $n=2 \cdot 3 \cdot 5 \cdots q+1$. As $n$ is not divisible by any prime $\le q$, either $n$ itself is a prime, or there exists a prime $>q$ which divides $n$. Both contradicts our initial assumption about prime numbers being finite. Over time, many algorithms have been formulated to count primes, some of them are very complex and require deeper understanding of number theory. So I shall not discuss about them in this post.
Prime numbers of special form have been studied in great details. Most famous of these special primes are Mersenne primes and Fermat primes.
Mersenne primes are named after Father Marin Mersenne, a French priest. Mersenne numbers are defined as $M_n=2^n-1 \hspace{1 mm}, n\geq 1$. It is not yet known whether the set of Mersenne primes are finite or infinite.
Famous mathematician Pierre de Fermat believed that numbers of form $2^{2^n}+1$ are primes, based on first $5$ values ($0\le n \le 4$). But no other Fermat primes are not known for $n>4$.
A lot of conjectures are there concerning primes. Most notable of them are Goldbach conjecture and twin prime conjecture. First one states that every even integer greater than $2$ can be expressed as the sum of two primes.
A couple of prime numbers have even been associated with superstitions. Most famous of them is the number $13$, considered as unlucky. Fear of number $13$ even has a special name, Triskaidekaphobia (it was mentioned in my favorite TV series Friends - S08E20 :) ). Other superstitious prime is the Belphegor's prime - $1000000000000066600000000000001$. It has exactly $31$ digits, inverse of $13$. It contains $666$, 'the number of the beast' at the center and have $13$ zeros on either sides. And it is also a palindromic prime.
Primes also have practical applications. Randomness of prime numbers and difficulty of factorizing lie at the heart of several public encryption schemes, which are used to secure communication systems.
Primes will most probably befuddle us for many more years to come. Where is the fun if the everything is predictable? :) As English novelist Mark Haddon said, "Prime numbers is what is left when you have taken all the patterns away."
Thanks for reading :)
Level level .... Chele ki diche 😂
ReplyDeleteTo analyze primes you have to prove the famous conjecture of Generalized Riemann Hypothesis. Also, not only mersenne primes but basically for any recurrence the questions of how many primes are there as a members of that recurrence is not solved. As a matter of fact, there is a conjecture that the number of fibonacci primes is finite. This is really interesting
ReplyDeleteI intentionally avoided Generalized Riemann Hypothesis, because it is too complicated and requires background knowledge. If we assume that Fermat primes are random, then expected number of Fermat primes is 2 / ln 2, but factors of Fermat numbers show some regularity. In any case, prime-related conjectures are extremely difficult to prove.
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