Is Two a Pseudo Prime?

Let me preface the following by saying that I am not a mathematician. I'm interested in math but only in the way that a lot of lunkheads are interested in NASCAR. That is, I'm a fan, but would only be in the way in gasoline alley.

With that disclaimer, I wanted to throw out the following thoughts for the consideration of the learned.

Two is a funny little number. It shows up everywhere in story and song and cuts a wide swath through human history. What am I saying? It cuts a wide swath through the history of everything and everybody of every description. One is the loneliest, but not two. Two rocks big time! Hardly an event happens where two isn't a prominent player. Two is even a prime number.

Or is it?

Primes are numbers which are evenly divisible only by themselves and by one.

They could be described as being aloof. They don't mix well with other numbers, at least not so far as mathematicians have been able to tell, with certainty. Mathematicians have been trying to nail down just what the relationship is between the primes and all the other numbers for a long time. It should be easy to find a formula for the nth prime but nobody has yet.

Getting back to two. Two is the only prime which is an even number. Putting it another way, in the very large set known as the prime numbers, two is the only one which is even. Odd isn't it? (Forgive the play on words.) I think so.

Some might say that fact constitutes grounds for removing two from the set of primes. I think so.

I think two is a pseudo, or accidental prime, a chameleon which takes on the appearance of being a prime by the accident of it's proximity to one. I don't think it belongs in the pattern of the primes at all.

I wonder if all the trouble mathematicians are having in discovering the pattern of the primes is, in some way, related to a mistaken assumption of the presence in that pattern of the number two.

Is two a pseudo prime?

It is a necessary factor in the factoral composition of even numbers so I say it's really the real deal.

reply to post by ipsedixit


From a conservative mathematical point of view, then yes, 2 is a prime number.
But having said that, the number 2 is unique out of the infinity of prime nmbers due to it being the ONLY prime number that is even while EVERY other prime number is odd.

If you'd like to learn more about prime numbers, then please read my thread "Adventures in Prime Number Land - A dummies Guide to Primes" at
www.abovetopsecret.com...

reply to post by EnlightenUp

It is the necessary factor which determines whether a number is even or not. In fact it is a factor which prevents fifty percent of all numbers from being considered primes. It interacts factorially with more numbers than any other number but one.

It is unlike all other prime numbers in its evenness.

One could say that there are two kinds of prime numbers, odd primes and even primes. Odd primes outnumber even primes by infinity to one.

That's what makes me wonder if two is a pseudo prime. A number which is a prime simply because, at that point in the number line it fufills the conditions of being a prime for a reason which is not the same as the reasons that determine whether other numbers will be primes.

Is it the Pluto of primes? Doomed to be reclassified.

reply to post by tauristercus

I don't know if my head can stand learning more about the primes!

I'm currently trying to read The Music of the Primes by Marcus du Sautoy, but thanks for the link. I'll take a look.

Originally posted by ipsedixit
I think two is a pseudo, or accidental prime, a chameleon which takes on the appearance of being a prime by the accident of it's proximity to one. I don't think it belongs in the pattern of the primes at all.

I wonder if all the trouble mathematicians are having in discovering the pattern of the primes is, in some way, related to a mistaken assumption of the presence in that pattern of the number two.

Is two a pseudo prime?

The principles of democracy apparently didn't make Two to leave the congregation of prime numbers.

Whether Two should or should not belong to the population of prime numbers is decided on a completetely different criterion than it's even/odd membership. There is a guy named Fundamental Theorem of Arithmetics -- a member of mathematical aristocracy -- to whom Two and other primes are subjects:
en.wikipedia.org...

Without this theorem, the field of number theory would collapse and become sacred numerology or something similar.

The FTA basically says that every integer greater than 1 is a unique product of primes. But if Two was excluded from "primership," the theorem would not be good for ALL even numbers. The multiplication of odd numbers -- no matter how many of them -- always returns odd number. Include just one even number into the multiplication and the result is even number. So imagine that odd numbers want to multiply and come up with a species that looks different -- like even. It's like the king has plenty of daughters but no son. So the odd numbers multiply and multiply, but there is no baby Even. So after almost infinite number of attempts, Two shows up and whenever it has sex with odd numbers, the kid is even.

And so think about number 2 as something quite powerful and extraordinary in the Kingdom of Primes. It's like a guy who wanders into a place where only women live: Wait! Girls, let's make it orderly . . . Please!!!! . . .


Number two is not a pseudoprime:
en.wikipedia.org...

reply to post by ipsedixit


There's also something peculiar about the relationship to 24 talked about by tauristercus (has this been proven for all p or is it just conjecture?) where 2 and 3 are not on the eight rays.

4! = 24 = 4*3*2*1 = 2^3 * 3 * 1

It almost seems to weave into spacetime itself and I get an eery feeling that I'm staring god in the face and can't see it.

The answer to everything will be related to permutations (factorials) representing all possible ways of choosing.

Sorry for the dab of metaphysical histrionics!

[edit on 9/29/2009 by EnlightenUp]

Originally posted by stander
Whether Two should or should not belong to the population of prime numbers is decided on a completetely different criterion than it's even/odd membership.

I've read through your post and quite enjoyed it. Perhaps I am wrong in this, but I thought that the only criterion for considering a number a prime is that it be evenly divisible uniquely by one and itself.

The FTA basically says that every integer greater than 1 is a unique product of primes.

To me that theorum adds an additional burden to the primes. Now it is not enough that they be uniquely evenly divisible by themselves and one only, but they are expected to generate all the other numbers too!

But here's the kicker. The only way they can do that is if one prime, two, does fifty percent of the work.

There is no question that two is a superstar, that it can do the "prime" trick, but it's eveness is odd. Makes me wonder if it is really a member of the club or just a clever interloper.

I don't have time to write more tonight, but I will check out your links.

reply to post by EnlightenUp

I'm turning in now EnlightenUp, but I will look at your post tomorrow.

Originally posted by ipsedixit
Perhaps I am wrong in this, but I thought that the only criterion for considering a number a prime is that it be evenly divisible uniquely by one and itself.

Well, if someone gives you a basket full of 1, 2, 3, 4, 5, ..., 99, 100 and asks you to take out those numbers that are divisible only by themselves and 1, then your pick would be made of some odd numbers. But number 2 would be included as well, coz it's property satisfies the condition:

2 / 2 = 1 (by itself)
2 / 1 = 2 (by 1)

It's like a place populated by millions of males and one female. You ask for a piano player, and those who can play the instrument step forward. The group would include that woman. Why would you exclude that woman from the group of piano players due to her unique gender, when she really can play piano, which stands for the condition "divisible by itself and 1?"

The problematic number would be number 1, coz

1 / 1 = 1 (by itself)
1 / 1 = 1 (by 1)

You can include number 1 into the population of primes -- until it starts to cause problems upon application. It's the Fundamental Theorem of Arithmetic where number 1 as a prime starts to cause trouble.

You can set 1+1=3 any time you please. But if you apply it as an architect and the roof collapses, it's time to try other options, like 1+1=2.

reply to post by ipsedixit


Is 2 a potential prime?

That depends on whether you use -2 or +2

Eg 1 + 1 = 2, most think this is correct till they are shown that there are 2 additional answers

(-1) + (-1) = -2
(+1) + (-1) = 0

HADES

reply to post by FTL_Navigator

It has nothing to do with whether you use -2 or +2. A prime number, by definition, is positive.

You seem to be claiming that negative one is identical to positive one. It is not.

But besides that, prime numbers have nothing to do with addition, 1 is not a prime, and 0 is not a prime.

reply to post by Phage

But besides that, prime numbers have nothing to do with addition ...

Sorry, that's incorrect as EVERY positive integer, and therefore EVERY prime number (which is a subset of the positive integers) can be derived using simple addition.

1 + 0 = 1
1 + 1 = 2 PRIME
2 + 1 = 3 PRIME
2 + 2 = 4
3 + 2 = 5 PRIME
3 + 3 = 6
4 + 3 = 7 PRIME
4 + 4 = 8
5 + 4 = 9
5 + 5 = 10
6 + 5 = 11 PRIME ....... and on to infinity (and beyond ! )

I realize that despite my questions about it, two is indeed a prime, just an odd one, being even. I don't know enough math, though to see how one is not considered to be a prime. Again, it satisfies the requirements of primes, doesn't it?

I thought that zero and infinity were the real trouble makers in mathematics. They are both considered "undefined" are they not? Therefore their usage is extremely restricted in mathematical operations, zero being confined to placeholder status in counting and infinity being confined to being equivalent to the phrase "and so on endlessly".

Math is a peculiar subject. I think it is best defined as God's way of playing ping pong with the brains of gifted materialists.

(Incidentally, during the course of following some links left by stander I came to realize that the words "pseudo prime" are used in a particular way in mathematical literature. Obviously the thread title has nothing to do with that usage.)

Edit: I'm reading tauristercus's thread on the primes and am informed that one is not considered to be a prime because mathematicians decided that primes must be evenly divisible, uniquely, by both themselves and one. One, surely the cleverest number in the whole number system, flouts that rule by leaving out one step. On that ground, and that ground alone (until I am informed otherwise), it is not considered to be a prime.

I can almost hear God cackling.

tauristercus, why don't you include numbers 2 and 3 as primes in your first diagram? I read that far and came to a screeching halt.

[atsimg]http://files.abovetopsecret.com/images/member/09a76a65da3c.jpg[/atsimg]


[edit on 30-9-2009 by ipsedixit]

Here is a graphic representation the first 500 prime numbers in binary.
I generated the picture with a couple lines of code in Mathematica.

[atsimg]http://files.abovetopsecret.com/images/member/b3fac035fea3.jpg[/atsimg]

reply to post by ipsedixit

tauristercus, why don't you include numbers 2 and 3 as primes in your first diagram? I read that far and came to a screeching halt.

As I mentioned in my opening prime, I am by no means even close to being what is defined as a "mathematician" but merely someone who has had a lifelong interest in primes and their seemingly random way of popping up like weeds amongst the set of positive integers.

Over the years, I've played around with them and eventually came to notice a very unusual pattern associated with primes, namely that the number 24 seems to play a PIVOTAL role amongst them.
Eventually I found that by creating an infinite series of concentric circles (as you saw in one of my diagrams) and subdividing each of those infinite number of circles into 24 sections, that once EVERY positive integer was plotted onto these circles, that EVERY prime except 2 and 3 would fall on ONLY 8 of those 24 sections. I eneded up calling those 8 sections "rays" as the primes would give the impression of shooting of into infinity in lines, or "rays".
Take another look at my diagrams and you'll quickly get the idea ... sometimes easier to see something then to have it explained

Finally, back to your question .... (about time, he says )

You'll notice that the number 1 appears on the 1st ray (ray1) and the next prime (5) appears on the ray after that (ray2). From this point on, no matter which prime you pick, it will only APPEAR on ray1 to ray8 ... NEVER anywhere else.

But where does that leave primes 2 and 3 ? Not on a ray, thats for sure !

To me that seemed really strange as except for these two primes, the remaining infinity of primes DO fall on these 8 rays. Also, 2 is the only prime out of the infinity of primes that's an EVEN number, so again, seems strange to me that 2 is afforded such a unique status !

Anyway, to get to the point, I'm looking at it from the point of view that perhaps we should actually be reinstating 1 back to primehood status (it is on a ray, after all) and perhaps demoting 2 and 3 to possible "pseudo prime" status.
I realize this is tantamount to mathematical sacrilege, but if mathematicians can demote 1 from it's original prime status (whoops, we made a mistake, sorry about that), then who's to say that we've actually been making the reverse error and according 2 and 3 prime status when in fact they're not and only "look" like primes.

Anyway, that's the way I'm looking at the primes and if you continue reading my thread, you'll see that no matter which way you turn, you're tripping over the number 24 or a multiple of 24 ! Notice also how EVERY prime appears capable of being created using ONLY multiples of 24

Originally posted by zerotensor
Here is a graphic representation the first 500 prime numbers in binary.
I generated the picture with a couple lines of code in Mathematica.

[atsimg]http://files.abovetopsecret.com/images/member/b3fac035fea3.jpg[/atsimg]

That's an interesting visual representation !

Can you please supply some kind of labelling to tell us what we're looking at ?

There is nothing unique about 2 or 3. Except that they are the smallest two primes. "No other prime is even" means the same as "no other prime (than 2) is a multiple of 2". But you can also say that no other prime than 17 is a multiple of 17. Nothing special.

Originally posted by nablator
There is nothing unique about 2 or 3. Except that they are the smallest two primes. "No other prime is even" means the same as "no other prime (than 2) is a multiple of 2". But you can also say that no other prime than 17 is a multiple of 17. Nothing special.

Not quite ....

By saying "no other prime (than 2) is a multiple of 2", you're effectively stating that out of the infinity of primes, that 2 is accorded a truly unique status by virtue of it being the only EVEN prime. Do you not stop to wonder WHY this number (2) doesn't EXACTLY fit the "mold" that the remaining infinity of primes are constructed from ?
Almost like saying "for every rule, there's an exception" ... as an explanation, simply not good enough.

I agree but did you read the rest of my post ? 17 is also unique for the same reason. Or 113. It's a general property of all primes, that no other prime is a multiple of them. Nothing special with 2 or 3. It is not an exception it is a rule. By definition all primes share this property.