Prime Numbers

[darnoconrad 0]

Hi guys,

I’d like to know more about prime numbers. I understand that they are natural numbers (1, 2, 3, etc…) that can only be divided by 1 and themselves. But apart from that I don’t know their significance and I’d like to get a better understanding about how to think about them.

Here are my questions about them:

Are there any interesting implications about them?

What are some common questions about them that are wrong… ie: How old is the universe?
Any properties one should always be considering when thinking about primes?

[Betsy Speicher 0]

Hi guys,

I’d like to know more about prime numbers. I understand that they are natural numbers (1, 2, 3, etc…) that can only be divided by 1 and themselves. But apart from that I don’t know their significance and I’d like to get a better understanding about how to think about them.

Here are my questions about them:

Are there any interesting implications about them?

What are some common questions about them that are wrong… ie: How old is the universe?

Any properties one should always be considering when thinking about primes?

An Objectivist named Marty Lewinter teaches number theory in college and he has a fun Facebook group called "Marty's Math Gang." We often discuss prime numbers and other interesting numerical ideas there. Check it out at https://www.facebook.com/groups/125183547645750/

[ewv 0]

To get a better understanding of how to think about them and the basic properties to keep in mind get a book on number theory. It's one branch of mathematics that to a large extent does not require a lot of preliminary knowledge of other areas of abstract mathematics. The site Betsy recommended can help you, but an inexpensive classic book published by Dover is Number Theory and its History by Ore (see chapter 4 -- so you don't have to read very far into the book to be able to understand the chapter on primes).

Interesting implications range from some fundamental and historical theorems of number theory, to famous unsolved problems, to their use in other areas of mathematics (like proving that √2 and more are irrational numbers as a very simple example), to the very practical applications today of prime numbers in cryptography.

They don't tell you how old the universe is. Mathematics is about method, not metaphysics.

Other than that you can search on the internet and hang around on the site Betsy recommended.

[ruveyn ben yosef 0]

To get a better understanding of how to think about them and the basic properties to keep in mind get a book on number theory. It's one branch of mathematics that to a large extent does not require a lot of preliminary knowledge of other areas of abstract mathematics. The site Betsy recommended can help you, but an inexpensive classic book published by Dover is Number Theory and its History by Ore (see chapter 4 -- so you don't have to read very far into the book to be able to understand the chapter on primes).

Interesting implications range from some fundamental and historical theorems of number theory, to famous unsolved problems, to their use in other areas of mathematics (like proving that √2 and more are irrational numbers as a very simple example), to the very practical applications today of prime numbers in cryptography.

They don't tell you how old the universe is. Mathematics is about method, not metaphysics.

Other than that you can search on the internet and hang around on the site Betsy recommended.

The most powerful results in number theory come from the analytic side. For that one must know the theory of real and complex variables.

Wiles resolution of "Fermat's Last Theorem" (WIth Fermat it was really a conjecture) involved the theory of modular groups.

Elementary number theory (the part of the theory that does not invoke methods for real and complex analysis) is indeed beautiful and a delight to the mathematical intellect. But the most important results ultimately from from Reimann's work of the Zeta Function

ruveyn

[ruveyn ben yosef 0]

Oops. I forgot to give a reference. Sorry about that.

http://en.wikipedia.org/wiki/Analytic_number_theory

Note to Ms. Moderator. Couldn't you give us 15 minutes to a a half our to edit our stuff?

ruveyn

[ewv 0]

The most powerful results in number theory come from the analytic side. For that one must know the theory of real and complex variables.

That's why I said to a "large extent". Most of the classical theory uses elementary methods. There is a lot he can learn about at that level.

[ewv 0]

Note to Ms. Moderator. Couldn't you give us 15 minutes to a a half our to edit our stuff?

You already have that. Use the preview, wait, and think functions. You don't even have to click for the last two. There is no time limit at all (other than normal lifespans).

[ruveyn ben yosef 0]

Note to Ms. Moderator. Couldn't you give us 15 minutes to a a half our to edit our stuff?

You already have that. Use the preview, wait, and think functions. You don't even have to click for the last two. There is no time limit at all (other than normal lifespans).

Good point. But what if the error is missed on the preview? Oh well. Being careful is a good idea. Thanks for the suggestion

ruveyn

[ewv 0]

Note to Ms. Moderator. Couldn't you give us 15 minutes to a a half our to edit our stuff?

You already have that. Use the preview, wait, and think functions. You don't even have to click for the last two. There is no time limit at all (other than normal lifespans).

Good point. But what if the error is missed on the preview? Oh well. Being careful is a good idea. Thanks for the suggestion

What if the error is missed if you had 15 minutes to re-edit? That's what the 'wait' and 'think' functions are for. You get enhanced results if you use the 'mull' plugin for the think function.

[darnoconrad 0]

An Objectivist named Marty Lewinter teaches number theory in college and he has a fun Facebook group called "Marty's Math Gang." We often discuss prime numbers and other interesting numerical ideas there. Check it out at https://www.facebook.com/groups/125183547645750/

Joined.

To get a better understanding of how to think about them and the basic properties to keep in mind get a book on number theory. It's one branch of mathematics that to a large extent does not require a lot of preliminary knowledge of other areas of abstract mathematics. The site Betsy recommended can help you, but an inexpensive classic book published by Dover is Number Theory and its History by Ore (see chapter 4 -- so you don't have to read very far into the book to be able to understand the chapter on primes).

Interesting implications range from some fundamental and historical theorems of number theory, to famous unsolved problems, to their use in other areas of mathematics (like proving that √2 and more are irrational numbers as a very simple example), to the very practical applications today of prime numbers in cryptography.

They don't tell you how old the universe is. Mathematics is about method, not metaphysics.

Other than that you can search on the internet and hang around on the site Betsy recommended.

I'll give the book a try on Saturday. Thanks.

But I was also hoping that we could post some interested bits on the forum.

[ewv 0]

I'll give the book a try on Saturday. Thanks.

But I was also hoping that we could post some interested bits on the forum.

You can get it from amazon here http://www.amazon.com/Number-Theory-History-Dover-Mathematics/dp/0486656209/ref=sr_1_1?ie=UTF8&qid=1363836336&sr=8-1&keywords=Number+Theory+and+its+History+Ore They have a kindle edition, too, but unfortunately the table of contents and other pages can't be sampled online at amazon for this book.

You might enjoy some related discussion (but not about prime numbers) here on the Forum starting after this post http://forums.4aynrandfans.com/index.php?showtopic=1678&page=2#entry13687 and at http://forums.4aynrandfans.com/index.php?showtopic=2649#entry21790

[darnoconrad 0]

Here are my thoughts so far about Prime Numbers:

Let me start with a definition of "Division"

- Division is separating a large group of uniform units into smaller uniform groups.

Thus, the meaning of "divisible only by 1 and itself" means...

- A Prime Number is a group of uniform units that cannot be separated into smaller uniform groups.

If one attempted to separate a prime into smaller uniform groups, one would find that there would remain one group that is different than all the others.

In the quest to discover the pattern for finding Prime Numbers one is asking what is common among all groups that cannot be divided into smaller uniform groups?

My question:

Is there necessarily a pattern? Which would mean there is a relationship between all Prime Numbers.

[ewv 0]

You "uniform groups" is ambiguous and irrelevant to the mathematics. A prime number is what you said it is in your first post: it is divisible only by itself and 1. Division in arithmetic is the inverse of and depends on the concept multiplication. Finding prime numbers means exactly finding numbers that can't be factored, i.e. can only be expressed as 1 times itself.

There is no known formula for computing the nth prime number. The number of primes up to a number n is approximately n/log(n) in the sense of a limit as n increases. This is shown in analytic number theory which is more technically advanced than most of what you will be reading at this point.

The famous unproved Goldbach conjecture says that every even number ≥ 6 is the sum of two odd primes. For more on "patterns" of primes look at mathematical sources like the Ore book. The conceptual and defining "relationship" between primes remains the fact that they can't be factored (except for the number itself and 1), but that is not expressed as a single computational formula.

Much of any branch of mathematics is the identification of mathematical relations, i.e., "patterns", which are established by technical means -- and technical expositions is the place to look for questions like this.

[ruveyn ben yosef 0]

You "uniform groups" is ambiguous and irrelevant to the mathematics. A prime number is what you said it is in your first post: it is divisible only by itself and 1. Division in arithmetic is the inverse of and depends on the concept multiplication. Finding prime numbers means exactly finding numbers that can't be factored, i.e. can only be expressed as 1 times itself.

There is no known formula for computing the nth prime number. The number of primes up to a number n is approximately n/log(n) in the sense of a limit as n increases. This is shown in analytic number theory which is more technically advanced than most of what you will be reading at this point.

The famous unproved Goldbach conjecture says that every even number ≥ 6 is the sum of two odd primes. For more on "patterns" of primes look at mathematical sources like the Ore book. The conceptual and defining "relationship" between primes remains the fact that they can't be factored (except for the number itself and 1), but that is not expressed as a single computational formula.

Much of any branch of mathematics is the identification of mathematical relations, i.e., "patterns", which are established by technical means -- and technical expositions is the place to look for questions like this.

Please have a look at this:

http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf

ruveyn

[ruveyn ben yosef 0]

You might also find this interesting:

http://arxiv.org/ftp/math/papers/0406/0406001.pdf

[darnoconrad 0]

You "uniform groups" is ambiguous and irrelevant to the mathematics. A prime number is what you said it is in your first post: it is divisible only by itself and 1. Division in arithmetic is the inverse of and depends on the concept multiplication.

My use of "uniform" is to be clear that I mean groups composed of an equal amount of units.

My use of "groups" is relevant because taking 6(units) and dividing by 3 means having 3 groups of 2 units.

example: IIIIII can be divided into two groups of units III and III

I'm trying to illustrate what one does when dividing.

Does division really depend on the concept of multiplication? Can you show why? I can certainly see how they are related as one is the inverse of the other, but I don't see why one has to come before the other?

The conceptual and defining "relationship" between primes remains the fact that they can't be factored (except for the number itself and 1), but that is not expressed as a single computational formula.

This is certainly a way of differentiating them against all other types of numbers, by describing this characteristic. But if we only look at prime numbers, is there another way they are related?

"In the quest to discover the pattern for finding Prime Numbers one is asking what is common among all groups that cannot be divided into smaller uniform groups?"

...because if there is a pattern to their reoccurrence then wouldn't that be considered a relationship between them?

[ewv 0]

You might also find this interesting:

http://arxiv.org/ftp/math/papers/0406/0406001.pdf

I'm not sure what in particular you mean to convey. They are interesting but (from looking them over briefly and assuming they are correct) don't seem to provide a closed form function in the way that is usually meant as a single formula of computation. They both count on sums of a 'characteristic' function that involves determining for each n in the sum whether or not n is prime, which is an algorithm not an arithmetic expression. One can always find the nth prime or the number of primes up to a given integer by finding and enumerating in order the primes up to that point, but there are more or less efficient ways of doing that.

[ewv 0]

Your "uniform groups" is ambiguous and irrelevant to the mathematics. A prime number is what you said it is in your first post: it is divisible only by itself and 1. Division in arithmetic is the inverse of and depends on the concept multiplication.

My use of "uniform" is to be clear that I mean groups composed of an equal amount of units.

My use of "groups" is relevant because taking 6(units) and dividing by 3 means having 3 groups of 2 units.

example: IIIIII can be divided into two groups of units III and III I'm trying to illustrate what one does when dividing.

Why are you trying to discuss arithmetic and prime number theory in terms of pre-mathematics scratching of tick marks instead of numbers? Mathematics is done using the concept of number, and relations -- such as simple laws of arithmetic -- between them. You can understand how the elementary concept of different numbers, and then the more general concept of number itself, arise in terms of repeated symbols of units, but if you want to think conceptually beyond that you need to use concepts of numbers routinely to build on. You don't keep repeating tick marks and overwhelming crows.

Elementary multiplication is repeated addition, and division is the inverse, repeated subtraction; you can visualize what that means in terms of "groups" of "units", but if you want to analyze prime numbers and the rest of mathematics, you had better do it in terms of simple mathematical concepts of arithmetic and algebra (at least), not the units on which they are based, if you expect to preserve mental unit economy through the use of abstractions from abstractions to avoid unnecessary or impossible complexity.

The Greeks did not have algebra and did not have the decimal number system. They managed to work with numbers geometrically, in part through arrays. They characterized square numbers, rectangle numbers, triangle numbers, etc., and the later Pythagoreans long before Euclid had the concept of prime numbers in such terms -- see the inexpensive Dover edition of Heath's 1931 Greek Mathematics. You might find that interesting, but to remain arrested at that level makes it much more difficult, and impossible to get to anything beyond the most elementary results.

Does division really depend on the concept of multiplication? Can you show why? I can certainly see how they are related as one is the inverse of the other, but I don't see why one has to come before the other?

At an elementary level of knowledge you could technically skip or ignore repeated addition and understand division as repeated subtraction, but what for? It's unlikely to arise that way chronologically and is a logical contortion in full context. Factorization as multiplication is essential for analyzing prime numbers.

The mathematical theory of the number systems is a formalization in an axiomatic approach of the laws of arithmetic that you first understand conceptually without it, for the purpose of systematically studying, expanding and validating the understanding you started with, and much more, in a hierarchical structure revealing logical dependencies and relationships, and as a basis for discovering and proving more advanced results you couldn't otherwise reach without that level of abstraction. (It is formulated abstractly algebraically and does not depend on any single system of units like the base 10 and decimal system, though of course you have to understand some such base system to get that far in the conceptual hierarchy.) In that system, addition logically precedes subtraction and multiplication logically precedes fractions and division.

See the classic and elementary Thurston, The Number-System and Landau, Foundations of Analysis.

The conceptual and defining "relationship" between primes remains the fact that they can't be factored (except for the number itself and 1), but that is not expressed as a single computational formula.

This is certainly a way of differentiating them against all other types of numbers, by describing this characteristic. But if we only look at prime numbers, is there another way they are related?

You couldn't understand prime numbers without first having the concept of number. 'Prime numbers' (like, e.g. fractions) are a subdivision of the concept 'number' at a higher level of abstraction (not to be confused with the formal structure in technical concepts like subsets and Cartesian cross products).

"In the quest to discover the pattern for finding Prime Numbers one is asking what is common among all groups that cannot be divided into smaller uniform groups?"

...because if there is a pattern to their reoccurrence then wouldn't that be considered a relationship between them?

There are many implicit relationships, but they exist in the context of numbers. The job of mathematics is to conceptualize and formulate them by discovering what is already implicit.