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An old mathematical puzzle soon to be unraveled?
phys.org ^ | 15 JAN 2014 | Benjamin Augereau

Posted on 01/21/2014 7:34:06 AM PST by onedoug

It is one the oldest mathematical problems in the world. Several centuries ago, the twin primes conjecture was formulated. As its name indicates, this hypothesis, which many science historians have attributed to the Greek mathematician Euclid, deals with prime numbers, those divisible only by themselves and by one (2, 3, 5, 7, 11, etc.). Under this assumption, there exists an infinite number of pairs of prime numbers whose difference is two, called twin primes (e.g., 3 and 5), but nobody has been able to confirm this so far.

(Excerpt) Read more at phys.org ...


TOPICS: History; Science
KEYWORDS: epigraphyandlanguage; euclid; godsgravesglyphs; greece; mathematics; primenumbers; twinprimesconjecture
I was the worst math student in my high school. Yet, as I grow older I find myself irresistibly drawn to number theory. I did take Calculus I/II in college, and so formed the basis of being largely self taught. I'd qualify that further by joking that I'm a theorist, and not an experimentalist.

But I really do love this stuff.

1 posted on 01/21/2014 7:34:06 AM PST by onedoug
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To: onedoug

Loving math and doing well in math class are two different things. With distance, all the things I learned in school are not only coming back to me, but I finally understand them.


2 posted on 01/21/2014 7:38:46 AM PST by RinaseaofDs
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To: onedoug

I think you may have meant “evenly divisible”. As far as proving an infinity, well good luck with that.


3 posted on 01/21/2014 7:39:02 AM PST by Dalberg-Acton
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To: onedoug
Gauß called number theory the “queen of mathematics.”
4 posted on 01/21/2014 7:42:58 AM PST by cartan
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To: onedoug
If you’ve had Calculus I and II in college, Hardy and Wright’s wonderful book An Introduction to the Theory of Numbers should be just right for you!
5 posted on 01/21/2014 7:46:21 AM PST by cartan
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To: Dalberg-Acton
As far as proving an infinity, well good luck with that.

I would suggest that if you stipulate that there are an infinite amount of 'prime numbers' at your disposal, then there are an infinite amount of pairs.

NOW. The article started off describing primes separated by '2'. Then it goes on to discuss primes separated by larger (Much larger) amounts.

IS the original problem about primes separated by '2', or primes separated by any random number?

I don't see the problem being quickly unraveled if it's the latter.

6 posted on 01/21/2014 7:55:29 AM PST by UCANSEE2 (I forgot what my tagline was supposed to say)
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To: UCANSEE2
I would suggest that if you stipulate that there are an infinite amount of 'prime numbers' at your disposal, then there are an infinite amount of pairs.
It is quite easy to see that there is an inifinitude of prime numbers: If you assume that there are only finitely many primes, and p1, …, pn are all prime numbers, then the product

p1×…×pn + 1

will be divisible by neither of these primes, and is bigger than any of the others, which leads to a contradiction! Hence, there are infinitely many primes. Unfortunately, this argument says absolutely nothing about the question whether there are infinitely many primes that are seperated by only 2 from another prime number!

NOW. The article started off describing primes separated by '2'. Then it goes on to discuss primes separated by larger (Much larger) amounts.

IS the original problem about primes separated by '2', or primes separated by any random number?

I don't see the problem being quickly unraveled if it's the latter.

The original problem is about primes seperated by 2. Proving it about primes being seperated by a little more is evidently simpler (but still very very hard!).
7 posted on 01/21/2014 8:09:26 AM PST by cartan
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To: cartan

Not in my library, and a little pricey on Amazon. I’ve been slogging about in Alan Baker’s “Intro...” book, but it gets pretty hard sometimes.

I really like what Paul Dirac once said: “It is more important to have beauty in one’s equations than to have them fit experiment.”

I’ve come to understand that where you find one, you usually find the other as well.

It’s difficult not to consider mathematics as having been imprinted on the universe.


8 posted on 01/21/2014 8:10:24 AM PST by onedoug
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To: UCANSEE2
The problem is proving it to infinity. Unless I misunderstand what the word “prove” means. Some things like 1/3 or 0.33333, I assume repeats infinitely but that's not the same as proving the prime pairs is it?
9 posted on 01/21/2014 8:13:57 AM PST by Dalberg-Acton
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To: onedoug

All this is not REAL science, anyway, just like “String Theory”, and Microbiology. Dr. Sheldon Cooper (he’s not crazy, his mom had him checked} solved this little riddle by the time he was 6, I’m sure.


10 posted on 01/21/2014 8:16:05 AM PST by jstaff
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To: onedoug
The concept of turning Δx into dx is pretty interesting, as pot #3 above indicates --
11 posted on 01/21/2014 8:16:53 AM PST by imardmd1 (Fiat Lux)
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To: onedoug

As an engineer, I have to point out that unless something can be proven and actually used, that is “have them fit the experiment”, we are in an intellectual wasteland. I could have phrased things more crudely (something about self-gratification), but chose not to.


12 posted on 01/21/2014 8:23:13 AM PST by Pecos (The Chicago Way: Kill the Constitution, one step at a time.)
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To: Dalberg-Acton

Prove it for 2 x infinity, and divide result by 2.


13 posted on 01/21/2014 8:26:27 AM PST by Scrambler Bob ( Concerning bo -- that refers to the president. If I capitalize it, I mean the dog.)
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To: Pecos

I can understand that. But I’m not an engineer, mathematician nor physicist, but just a dabbler excited by what I see as evidence of God in various results.


14 posted on 01/21/2014 8:32:37 AM PST by onedoug
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To: onedoug
EVERY number is divisible by one, and EVERY even number is divisible by two.

Therefore Prime Numbers must be odd and ... since there is an infinite number of odd numbers, some of which are prime ... it follows that there must be a relatively infinite number of odd prime numbers because infinity is infinity, and infinity is without end.

QED.

15 posted on 01/21/2014 8:33:01 AM PST by OldNavyVet (Robert's revenge lies in our getting rid of Democrat control in House, Senate and Preisdent.)
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To: onedoug
Here's one that will even interest kids. It's called the Collatz Conjecture.

You may start with any number.

   If the number is odd, multiply it by 3 then add 1.

   if the number is even, divide by two.

Take the number you end up with and repeat. Eventually you will end up with the number 1.

I wrote a script to play with this a bit. Here are a few examples of the output:

$ collatz 10
10 5 16 8 4 2 1
Starting with the initial value of 10, it takes 6 steps to reach unity.
$ collatz 25
25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
Starting with the initial value of 25, it takes 23 steps to reach unity.
$ collatz 26
26 13 40 20 10 5 16 8 4 2 1
Starting with the initial value of 26, it takes 10 steps to reach unity.
$ collatz 27
27 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
Starting with the initial value of 27, it takes 111 steps to reach unity.
$ collatz 28
28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
Starting with the initial value of 28, it takes 18 steps to reach unity.
$ collatz 29
29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
Starting with the initial value of 29, it takes 18 steps to reach unity.
 

Look up the wiki entry for the Collatz Conjecture. It has some interesting stuff on plotting the number of steps as well. Here's a variation, where we're not interested in the numbers returned, but instead are interested in the number of steps and the highest number reached in the series. For some reason 9232 is a really common mumber you see as the highest number, which you'll find if let the program chomp on a few thousand numbers

$ collatz2 2 40 | grep highest
2 takes 1 steps, and 2 is the highest number reached.
3 takes 7 steps, and 16 is the highest number reached.
4 takes 2 steps, and 4 is the highest number reached.
5 takes 5 steps, and 16 is the highest number reached.
6 takes 8 steps, and 16 is the highest number reached.
7 takes 16 steps, and 52 is the highest number reached.
8 takes 3 steps, and 8 is the highest number reached.
9 takes 19 steps, and 52 is the highest number reached.
10 takes 6 steps, and 16 is the highest number reached.
11 takes 14 steps, and 52 is the highest number reached.
12 takes 9 steps, and 16 is the highest number reached.
13 takes 9 steps, and 40 is the highest number reached.
14 takes 17 steps, and 52 is the highest number reached.
15 takes 17 steps, and 160 is the highest number reached.
16 takes 4 steps, and 16 is the highest number reached.
17 takes 12 steps, and 52 is the highest number reached.
18 takes 20 steps, and 52 is the highest number reached.
19 takes 20 steps, and 88 is the highest number reached.
20 takes 7 steps, and 20 is the highest number reached.
21 takes 7 steps, and 64 is the highest number reached.
22 takes 15 steps, and 52 is the highest number reached.
23 takes 15 steps, and 160 is the highest number reached.
24 takes 10 steps, and 24 is the highest number reached.
25 takes 23 steps, and 88 is the highest number reached.
26 takes 10 steps, and 40 is the highest number reached.
27 takes 111 steps, and 9232 is the highest number reached.
28 takes 18 steps, and 52 is the highest number reached.
29 takes 18 steps, and 88 is the highest number reached.
30 takes 18 steps, and 160 is the highest number reached.
31 takes 106 steps, and 9232 is the highest number reached.
32 takes 5 steps, and 32 is the highest number reached.
33 takes 26 steps, and 100 is the highest number reached.
34 takes 13 steps, and 52 is the highest number reached.
35 takes 13 steps, and 160 is the highest number reached.
36 takes 21 steps, and 52 is the highest number reached.
37 takes 21 steps, and 112 is the highest number reached.
38 takes 21 steps, and 88 is the highest number reached.
39 takes 34 steps, and 304 is the highest number reached.
40 takes 8 steps, and 40 is the highest number reached.
 

Fun with numbers!

16 posted on 01/21/2014 8:39:41 AM PST by zeugma (Is it evil of me to teach my bird to say "here kitty, kitty"?)
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To: zeugma

Indeed.


17 posted on 01/21/2014 8:42:06 AM PST by onedoug
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To: Dalberg-Acton

If you can prove that the number can’t be finite, then you will have proven it must be infinite.


18 posted on 01/21/2014 8:47:51 AM PST by Boogieman
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To: onedoug

Sieve of Eratosthenes

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


19 posted on 01/21/2014 9:03:58 AM PST by Hostage (ARTICLE V)
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To: Pecos

I could have phrased things more crudely (something about self-gratification), but chose not to.

(((
This lady appreciates that and prays that your sense of gentlemanly behavior spreads on FR.


20 posted on 01/21/2014 9:30:53 AM PST by Bigg Red (O LORD, our Lord, how majestic is your name in all the earth! Ps 8)
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To: OldNavyVet
EVERY number is divisible by one, and EVERY even number is divisible by two.

Therefore Prime Numbers must be odd and ... since there is an infinite number of odd numbers, some of which are prime ... it follows that there must be a relatively infinite number of odd prime numbers because infinity is infinity, and infinity is without end.

QED.

What about 2? Isn't it prime?

21 posted on 01/21/2014 10:27:53 AM PST by OneWingedShark (Q: Why am I here? A: To do Justly, to love mercy, and to walk humbly with my God.)
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To: OneWingedShark
What about 2? Isn't it prime?

You're right. Two is a prime number. My thinking was infinitely out of sight.

22 posted on 01/21/2014 10:32:57 AM PST by OldNavyVet (Robert's revenge lies in our getting rid of Democrat control in House, Senate and Preisdent.)
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To: zeugma

Do you suppose there might be a finite number of highest numbers?


23 posted on 01/21/2014 11:03:10 AM PST by Marylander
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To: Marylander
Do you suppose there might be a finite number of highest numbers?

No, but I have found after running several million tests, that there are major clusters similar to 9232 found in higher regions. One thing that I think is cool is that with the exception of numbers that are powers of 2, (powers of 2 are always the shortest route to 1 for obvious reasons)  there is no easy way to guess how many steps a given number will take to get to 1 without running it.

From the list I posted previously...

25 takes 23 steps, and 88 is the highest number reached.
26 takes 10 steps, and 40 is the highest number reached.
27 takes 111 steps, and 9232 is the highest number reached.
28 takes 18 steps, and 52 is the highest number reached.
29 takes 18 steps, and 88 is the highest number reached.
 

The number 27 is definitely an outlier in the number of required steps.

There are other variations on the Collatz Conjecture. One multiplies by 5 and divides by 2, and has the same result. Strangely that 9232 number pops up there and in other variations as well. I did some searching on the 9232 number and couldn't find anything specific about it.

I wish I'd known about stuff like this when I was young and stupid.  

24 posted on 01/21/2014 11:21:25 AM PST by zeugma (Is it evil of me to teach my bird to say "here kitty, kitty"?)
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To: zeugma
Kinda bored here, so I thought I'd follow up on the 9232 number. To quantify how often that particular number shows up, lets look at the numbers between 2 and 5000. What percentage of those numbers have 9232 as a part of their 'path' on the Collatz number tree? To find out, I used a bash script that I wrote that will take as input the number to start and stop on.  Unfortunately, for my purpose, it also prints out a line that says what the highest number reached was, so we have to suppress it. That's what the "grep -v reached" is for.

$ collatz2 2 5000 | grep 9232 | grep -v reached | wc -l
1996
 

So out of 5000 numbers, 1996, or  39.9% of them will pass through the number 9232.

Yup. That's a lot. But is it unusual? Well let's take a look at the numbers that surround it...

$ for x in 1 2 3 4 5 6 7 8 9; do
> collatz2 2 5000 | grep 923${x} | grep -v reached | wc -l
> done
0
1996
8
23
0
0
0
2
0
 

So, looking at the above, that translates to the following

9231 = 0
9232 = 1996
9233 =8
9234 =23
9235 =0
9236 =0
9237 =0
9238 =2
9239 =0

I'd call that unusual. What does it mean, if anything? I have no idea.

 

25 posted on 01/21/2014 12:12:17 PM PST by zeugma (Is it evil of me to teach my bird to say "here kitty, kitty"?)
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To: StayAt HomeMother; Ernest_at_the_Beach; decimon; 1010RD; 21twelve; 24Karet; 2ndDivisionVet; ...
Thanks onedoug.
...the twin primes conjecture... which many science historians have attributed to the Greek mathematician Euclid, deals with prime numbers...

26 posted on 01/21/2014 5:19:19 PM PST by SunkenCiv (;http://www.freerepublic.com/~mestamachine/)
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I think the Euclid connection was discovered by Marty McFly when he went back in time from the parking lot of the Twin Primes Mall.


27 posted on 01/21/2014 5:26:47 PM PST by SunkenCiv (;http://www.freerepublic.com/~mestamachine/)
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http://polymathprojects.org/

http://www.polymath-software.com/


28 posted on 01/21/2014 5:30:56 PM PST by SunkenCiv (;http://www.freerepublic.com/~mestamachine/)
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http://primes.utm.edu/


29 posted on 01/21/2014 5:32:00 PM PST by SunkenCiv (;http://www.freerepublic.com/~mestamachine/)
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https://www.google.com/search?q=Collatz+Conjecture


30 posted on 01/21/2014 5:33:33 PM PST by SunkenCiv (;http://www.freerepublic.com/~mestamachine/)
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To: onedoug; a fool in paradise; Slings and Arrows

31 posted on 01/21/2014 5:35:16 PM PST by Revolting cat! (Bad things are wrong! Ice cream is delicious! We reserve the right to serve refuse to anyone!)
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To: onedoug

The answer is 42.

(HHGTTG)


32 posted on 01/21/2014 6:11:36 PM PST by Redcitizen (Never bring a tank to a Chuck Norris fight.)
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To: onedoug
...as I grow older I find myself irresistibly drawn to number theory.

I'm much the same. Have you read "Fermat's Enigma"?

I think you'd enjoy it.

33 posted on 01/21/2014 6:56:15 PM PST by Straight Vermonter (Posting from deep behind the Maple Curtain)
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To: onedoug
I believe that everything we see as natural laws are manifestations of a single universal Truth. Any conflicts we “see” are due to our inability to be attuned to Him.
34 posted on 01/22/2014 5:25:54 AM PST by Pecos (The Chicago Way: Kill the Constitution, one step at a time.)
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To: Straight Vermonter

Read it some years ago.

Yutaka Tanyama was a genius for the ages.


35 posted on 01/22/2014 5:46:04 AM PST by onedoug
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