An old mathematical puzzle soon to be unraveled?

phys.org ^ | 15 JAN 2014 | Benjamin Augereau

[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.

[onedoug]

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.

[RinaseaofDs]

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.

[Dalberg-Acton]

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

[cartan]

Gauß called number theory the “queen of mathematics.”

[cartan]

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!

[UCANSEE2]

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.

[cartan]

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, …, p_n are all prime numbers, then the product

p₁×…×p_n + 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!).

[onedoug]

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.

[Dalberg-Acton]

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?

[jstaff]

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.

[imardmd1]

The concept of turning Δx into dx is pretty interesting, as pot #3 above indicates --

[Pecos]

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.

[Scrambler Bob]

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

[onedoug]

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.

[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.

[zeugma]

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!

[onedoug]

Indeed.

[Boogieman]

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

[Hostage]

Sieve of Eratosthenes

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

[Bigg Red]

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.