PBS Discussions

NOVA scienceNOW1 · Moderator Joined: 13 Jan 2005 Posts: 64

In 2005, mathematicians Dan Goldston of San Jose State University, Cem Yalcin Yildrim of Istanbul’s Bogazici University, and Janos Pintz of the Hungarian Academy of Science wrote a new proof that may answer one of the most persistent math problems of all time—whether or not there are an infinite number of so-called “twin primes.” Twin primes are pairs of prime numbers that occur as p and p + 2 such as 3 and 5 or 11 and 13. Do prime numbers hold particular interest for you? Are there other elusive math solutions you are waiting to be uncovered? What is the Holy Grail of mathematics in your mind?

timquinn · Joined: 10 Jan 2006 Posts: 1 Location: Los Angeles

Great song, twin prime conjecture song. Did you know there is already a well known twin prime song? yes indeed, the hit song from a few years ago, 8675309, ostensibly Jennys' phone number, by Tommy Tutone (I looked it up) is named after the smaller of a twin prime pair. Could not be an accident.

I will leave it for the student to figure out its pair.
_________________
Tim Quinn
timquinn.nu
dangerouscurve.org
arsfidelis.org

brewma · Joined: 11 Jan 2006 Posts: 6

i really like the song and it opend my mind up to the math field, i always thought that it was a some what boring subject but that song kind of made it fun

Anubis · Joined: 11 Jan 2006 Posts: 2

Where's 23's twin?
Wouldn't it be more revealing to map out the non-twinned primes?
When I was younger my mother once told me she had a hard time falling asleep at nights. I explained to her the concept of prime numbers and told her to count them until she fell asleep. She turns '80 on 1/20. Wonder how high she's counted? Smile

cedialogs · Posted: Wed Jan 11, 2006 11:24 pm Post subject: Re: Twin Prime Conjecture

I think I solved the "twin primes problem" about 35 years ago. It was not done in the context of mathematical research, but as an exercise in building a CPU loading exercise for benchmarking processors. Each subsequent refinement to the program over the years was done as psychotherapy whenever I needed to divert my mind from the difficulties of the moment.

When working on the original FORTRAN version of the program the twin primes phenomenon became apparent. All prime numbers after three are of the form 6n-1 or 6n+1 where n is any natural number. 6n+2 and 6n+4 are even, and hence not prime. 6n+3 is always divisible by 3. 6n+5 is the same number as 6n-1 in that the former is just one iteration further into the series than the latter.

Either 6n+1 or 6n-1 can be non-prime for some values of n, but all primes after 3, whether paired or not, must be of this form. Primes become less frequent among progressively larger numbers, and the average distance between prime numbers appears to increase in proportion to something like the square root of the location in the number sequence. For example, the sparseness of primes near 10,000 appears to be somewhere around 10 times the sparseness around 100. This increasing sparseness is shown by my prime numbers benchmark program, which reports that among all natural numbers less than 100,000,000 the longest range of continuously factorable numbers (i.e. non-prime numbers) contains 219 integers. It extends from 47,326,694 through 47,326,912. Interestingly, this range occurs closer to the start of the list than to the list's end.

The remaining question is whether there are an infinite number of paired primes of the form 6n-1 and 6n+1. It would appear that as n becomes arbitrarily large the number of paired primes will become progressively more sparse, but never non-existent. Considering that other participants in this forum are contending that proving that the number of these paired primes is infinite is quite complex, I will not attempt a complete, formal proof as I have other things to do with my life.

However, for those more obsessed with this topic than I, I would suggest that there should be a simple proof of the twin primes conjecture based on the algorithm used in the Sieve of Eratosthenes. Recognizing that all primes after 3 must be of the form 6n-1 or 6n+1, one sees that all that the sieving process does is kill off either one or both of the numbers defined by this formula.

It should be possible through mathematical induction to demonstrate that on some subsequent executions of the sieve's inner loop some, but not all, of the pairs between its starting index and infinity will be killed. It should also be possible to show that later executions of the inner loop would from time to time leave behind one or more proven prime pairs. As long as this infinite process continues to leave valid prime pairs behind, the number of these pairs is infinite.

Formalizing this exercise should produce a highly cost effective and air tight proof of the twin primes hypothesis since it would require a tiny fraction of the effort that has apparently been put into the project to date. It really shouldn't be all that difficult considering that Eratosthenes and I, in our infinite modesty, have already done most of the work.

If anyone is interested in a copy of the current C# prime number calculating benchmark, contact me. The system requirements for the program are a Microsoft Windows system WITH THE DOT NET FRAMEWORK INSTALLED. Free memory required to calculate all prime numbers less than 1,000,000,000 is about 140MB, with less memory required for a smaller upper limit. This single threaded benchmark calculates all primes less than 100,000,000 in 2.8 seconds when using an Athlon 64 3200+ processor.

goldston · Joined: 12 Jan 2006 Posts: 1 Location: San Jose, CA

I liked the song too. I would like to offer a few clarifications to the show. The most important is as stated in the first message here but not made clear on the show, this new result is joint work of Cem Yildirim, Janos Pintz, and me. Cem and I have been working together on this problem since 1999; Pintz made a decisive contribution a year ago and the three of us have been working on this problem together since then. To say it is Goldston's discovery is wrong. We refer to it as the GPY result, and this seems to becoming the standard way to refer to the work.

I am actually at San Jose State University, not the U of San Jose, but I guess it is hard to fit that into a song.

No one really knows if Euclid made the twin prime conjecture. He does have a proof that there are infinitely many primes, and he or other Greeks could easily have thought of this problem, but the first published statement seems to be due to de Polignac in 1849. Strangely enough, the Goldbach conjecture that every even number is a sum of two primes seems less natural but was conjectured about 100 years before this.

Finally, the GPY team and many other mathematicians are now putting a lot of effort into proving that there will always be pairs of primes a bound distance apart, which can be viewed as a big step towards proving the twin prime conjecture. It is unclear whether we are going to succeed anytime soon, but up until a year ago I would have sworn that this problems was beyond our 21st century mathemetics to resolve. No more!

Wilson · Joined: 12 Jan 2006 Posts: 1 Location: University of Oregon

Primes are NOT "numbers divisible by themselves and 1".

...and that means, contrary the list of primes shown on the screen, 1 is not prime.

Why should we care? Because the most important theorem (with all respect to Goldston's work) on prime number is the one every grammer school student knows:

NOVA scienceNOW1 · Moderator Joined: 13 Jan 2005 Posts: 64

We greatly appreciate Dr. Goldston’s reaction to the musical interlude in episode 5 of NOVA scienceNOW. Drs. Yildrem and Pintz were partners with Goldston and their omission from the song was a decision based on musicality – not merit. This segment was designed primarily to entertain and teach the concept of Twin Prime Conjecture by relating it to the new proof. Unfortunately, due to the short segment time, we did not provide a complete review of the topic. As to Euclid’s relationship to Twin Primes, there is debate over whether he recognized them first, but it is fair to say, as Dr. Goldston has pointed out, that he did not explicitly formulate the conjecture in its modern form. Euclid was, however, extremely interested in prime numbers and laid the foundation for all later work in this field. We sincerely regret the error on the name of Dr. Goldston's academic affiliation. For more information on the history of the Twin Prime Conjecture, Euclid and the GPY proof, we direct interested readers to our links & books at http://www.pbs.org/wgbh/nova/sciencenow/3302/resources.html#h02

-The NOVA scienceNOW production team

gary beachum · Joined: 19 Jan 2006 Posts: 1

I think it is remarkable that there have been no (or very few) uniform pieces of metal or stone, such as coins, or other forms of readily usable currency found among the artifacts of any pre-columbian civilization. Certainly there was extensive metal working, and gold, silver, and copper appear to have been valued, but why no money? If they were keeping transferable records of accounts--the ubiquitous Myan knotted ropes,--then they had an economy that depended exclusively on M2. Very impressive! The thing about the knotted ropes that's relevant to this topic is that one solution to this system (which is consistent with the Myan written counting system) is that it was a progessive base number system. That is, one, then base 2, then base 4, then base 20 (zero was used as a place holder). In other words, they had no primes. Why?! And why did the ropes never go past 157? (I may have some of the details wrong, but the gist is correct.)
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Gary Beachum

Richard Granvold · Joined: 08 Feb 2007 Posts: 2 Location: Albany Oregon

To know the number of primes is to know the number of twin primes. I discovered this in my research on the Twin Primes Proof. Here it is.
P_2(n) = summation from 1 to N of INT{[p(n) - P(n - 6)]/2}
where n = 6(N) + 1. As far as I know this is a new relation. This is based on the prime differrence of 6. I have more information on this as it related to the proof of the twin primes conjecture. Try it and have fun. Thank you!

Richard Granvold · Joined: 08 Feb 2007 Posts: 2 Location: Albany Oregon

Because so many have viewed my last thoughts I would like to offer some mor insights as to why there are, I believe, an infinity of twin primes.
1.) The prime numbers and therefore the twin primes after3, (3, 5) are next to whole multiples of 6.
2.) This being true the potential primes and twin primes after 3, (3,5), form two columns of numbers those one less than the multiples of 6 and those one more than the multiples of 6 and since there are an infinity of primes these columns of potential primes have an infinity of rows.
3.) If to each and every row of these two columns of potential primes more than a columns worth are prime then there is at least one row with two prime numbers next to the same multiple of 6 and thus are twin primes.
4. Via the sieve of Erastostenes a sieve count series can be constructed for each column which in term to term comparision with the series expansion of ln2 - 1/2 times the number of rows being sieved indicates that this is indeed the case. There are an infinity of twim primes.
My research has led my to the above. I have a detailed written proof of the above. Howevery after having sent the proof to the journals I have found that my proof in not being read!! No one believes that a proof has been constructed especially by a nobody so they won't even look at it.
Alas! What is to be done? Thank you and enjoy.

common sense · Joined: 09 Jul 2007 Posts: 6

But what is P(n-6) in the expression? How do you calculate it?

My e-mail is w1234s1234@yahoo.com