Prime Numbers

  • #1
--> Why are prime numbers so important to number theory? (Apart from speculations of being connected to energy levels of complex quantum systems.)

--> Let for time being, primes that we know of, be called primes of "type-2". Here '2' comes from the definition of primes. Since we consider primes as those numbers, which have no "pair" of factors, apart from 1 and itself.
If a number c is not prime, then c= a*b has atleast some a and b ≠1 or c.

--> Let us now call a number, a prime of "type-3",such that number has no 3-tuple of factors apart from 1 or number itself.
It's converse can be stated as:
d = a*b*c
where a,b,c ≠ 1or d.

--> it can be observed that primes of 'type 2' are actually subset of 'type 3'.
--> All 2-digit numbers are primes of type 3.
--> If we go on increasing our type numbers, all smaller types are actually subset of larger types.

--> Doesn't this makes idea of primes look abstract or random, and defining them on basis of 'pairs of factors' a kind-of bias?
--> My apolgies, if i said something wrong.
 

Answers and Replies

  • #2
695
2
Hi, sciencecrazy,
primes, as it is often said, are the 'building blocks' of natural numbers (the numbers you use to count: 1, 2, 3, 4, ...). The natural numbers are "bags of primes": you can imagine each one as a little bag containing a collection of primes (possibly repeated) inside. The content of this bag is a kind of "signature", personal and unique for each number: the number 20, for example, has two 2's and one 5 in its bag, and it is the only number with this content; another number would have a different collection of primes in its bag. The number 1 is the owner of the empty bag, with no primes in it. Many properties in number theory are traceable to these "bags": for example, two numbers are coprime when the intersection of their bags is empty.

As for your "type-n" classification, consider the differences between types. For example, think of the "type-3" primes which are *not* of "type-2". These are numbers with exactly two primes in their bags, neither more nor less. They are known as "semiprimes", and they play a role in cryptography. The principle is that, for very large numbers, multiplying two primes to produce a semiprime is very easy; while, on the other hand, having the semiprime and trying to find the two primes that compose it is computationally very expensive.
 
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