Exploring Primes: Is 'Pair of Factors' Bias?

In summary: Therefore, encryption systems based on the multiplication of large semiprimes are considered very secure.In summary, primes are important in number theory because they are the building blocks of natural numbers and have unique properties that can be traced back to these "bags of primes". They also have practical applications, such as in cryptography, making them even more significant. The classification of primes into "type-n" categories reveals their distinct characteristics and further demonstrates their significance in mathematics.
  • #1
sciencecrazy
5
0
--> 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.
 
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  • #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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1. What are primes and why are they important?

Primes are numbers that are only divisible by 1 and themselves. They are important in cryptography, number theory, and other fields of mathematics.

2. What is the "pair of factors" bias and how does it relate to primes?

The "pair of factors" bias is a phenomenon where people tend to focus on the factors of a number that are close together, rather than considering all possible factors. This bias can lead to incorrect assumptions and conclusions about the properties of primes.

3. How does exploring primes help us better understand mathematics?

Exploring primes allows us to uncover patterns and relationships between numbers, which can lead to deeper understanding of mathematical concepts. It also helps us develop problem-solving skills and critical thinking abilities.

4. What are some methods for exploring primes and overcoming bias?

There are several methods for exploring primes, such as creating tables and graphs, using modular arithmetic, and applying various mathematical theorems. To overcome bias, it is important to consider all factors of a number and not just the ones that are convenient or easy to work with.

5. Can exploring primes have practical applications?

Yes, exploring primes has many practical applications, particularly in the field of cryptography. Primes are used in encryption algorithms to ensure secure communication and protect sensitive information. They also have applications in fields such as computer science and physics.

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