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Prime Numbers Get Hitched
Seed Magazine ^ | Feb/Mar 2006 | Marcus du Sautoy

Posted on 04/11/2006 3:08:56 PM PDT by LibWhacker

In their search for patterns, mathematicians have uncovered unlikely connections between prime numbers and quantum physics. Will the subatomic world help reveal the elusive nature of the primes?

In 1972, the physicist Freeman Dyson wrote an article called "Missed Opportunities." In it, he describes how relativity could have been discovered many years before Einstein announced his findings if mathematicians in places like Göttingen had spoken to physicists who were poring over Maxwell's equations describing electromagnetism. The ingredients were there in 1865 to make the breakthrough—only announced by Einstein some 40 years later.

It is striking that Dyson should have written about scientific ships passing in the night. Shortly after he published the piece, he was responsible for an abrupt collision between physics and mathematics that produced one of the most remarkable scientific ideas of the last half century: that quantum physics and prime numbers are inextricably linked.

This unexpected connection with physics has given us a glimpse of the mathematics that might, ultimately, reveal the secret of these enigmatic numbers. At first the link seemed rather tenuous. But the important role played by the number 42 has recently persuaded even the deepest skeptics that the subatomic world might hold the key to one of the greatest unsolved problems in mathematics.

Prime numbers, such as 17 and 23, are those that can only be divided by themselves and one. They are the most important objects in mathematics because, as the ancient Greeks discovered, they are the building blocks of all numbers—any of which can be broken down into a product of primes. (For example, 105 = 3 x 5 x 7.) They are the hydrogen and oxygen of the world of mathematics, the atoms of arithmetic. They also represent one of the greatest challenges in mathematics.

As a mathematician, I've dedicated my life to trying to find patterns, structure and logic in the apparent chaos that surrounds me. Yet this science of patterns seems to be built from a set of numbers which have no logic to them at all. The primes look more like a set of lottery ticket numbers than a sequence generated by some simple formula or law.

For 2,000 years the problem of the pattern of the primes—or the lack thereof—has been like a magnet, drawing in perplexed mathematicians. Among them was Bernhard Riemann who, in 1859, the same year Darwin published his theory of evolution, put forward an equally-revolutionary thesis for the origin of the primes. Riemann was the mathematician in Göttingen responsible for creating the geometry that would become the foundation for Einstein's great breakthrough. But it wasn't only relativity that his theory would unlock.

Riemann discovered a geometric landscape, the contours of which held the secret to the way primes are distributed through the universe of numbers. He realized that he could use something called the zeta function to build a landscape where the peaks and troughs in a three-dimensional graph correspond to the outputs of the function. The zeta function provided a bridge between the primes and the world of geometry. As Riemann explored the significance of this new landscape, he realized that the places where the zeta function outputs zero (which correspond to the troughs, or places where the landscape dips to sea-level) hold crucial information about the nature of the primes. Mathematicians call these significant places the zeros.

Riemann's discovery was as revolutionary as Einstein's realization that E=mc2. Instead of matter turning into energy, Riemann's equation transformed the primes into points at sea-level in the zeta landscape. But then Riemann noticed that it did something even more incredible. As he marked the locations of the first 10 zeros, a rather amazing pattern began to emerge. The zeros weren't scattered all over; they seemed to be running in a straight line through the landscape. Riemann couldn't believe this was just a coincidence. He proposed that all the zeros, infinitely many of them, would be sitting on this critical line—a conjecture that has become known as the Riemann Hypothesis.

But what did this amazing pattern mean for the primes? If Riemann's discovery was right, it would imply that nature had distributed the primes as fairly as possible. It would mean that the primes behave rather like the random molecules of gas in a room: Although you might not know quite where each molecule is, you can be sure that there won't be a vacuum at one corner and a concentration of molecules at the other.

For mathematicians, Riemann's prediction about the distribution of primes has been very powerful. If true, it would imply the viability of thousands of other theorems, including several of my own, which have had to assume the validity of Riemann's Hypothesis to make further progress. But despite nearly 150 years of effort, no one has been able to confirm that all the zeros really do line up as he predicted.

It was a chance meeting between physicist Freeman Dyson and number theorist Hugh Montgomery in 1972, over tea at Princeton's Institute for Advanced Study, that revealed a stunning new connection in the story of the primes—one that might finally provide a clue about how to navigate Riemann's landscape. They discovered that if you compare a strip of zeros from Riemann's critical line to the experimentally recorded energy levels in the nucleus of a large atom like erbium, the 68th atom in the periodic table of elements, the two are uncannily similar.

It seemed the patterns Montgomery was predicting for the way zeros were distributed on Riemann's critical line were the same as those predicted by quantum physicists for energy levels in the nucleus of heavy atoms. The implications of a connection were immense: If one could understand the mathematics describing the structure of the atomic nucleus in quantum physics, maybe the same math could solve the Riemann Hypothesis.

Mathematicians were skeptical. Though mathematics has often served physicists—Einstein, for instance—they wondered whether physics could really answer hard-core problems in number theory. So in 1996, Peter Sarnak at Princeton threw down the gauntlet and challenged physicists to tell the mathematicians something they didn't know about primes. Recently, Jon Keating and Nina Snaith, of Bristol, duely obliged.

There is an important sequence of numbers called "the moments of the Riemann zeta function." Although we know abstractly how to define it, mathematicians have had great difficulty explicitly calculating the numbers in the sequence. We have known since the 1920s that the first two numbers are 1 and 2, but it wasn't until a few years ago that mathematicians conjectured that the third number in the sequence may be 42—a figure greatly significant to those well-versed in The Hitchhiker's Guide to the Galaxy.

It would also prove to be significant in confirming the connection between primes and quantum physics. Using the connection, Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence. Prior to this breakthrough, the evidence for a connection between quantum physics and the primes was based solely on interesting statistical comparisons. But mathematicians are very suspicious of statistics. We like things to be exact. Keating and Snaith had used physics to make a very precise prediction that left no room for the power of statistics to see patterns where there are none.

Mathematicians are now convinced. That chance meeting in the common room in Princeton resulted in one of the most exciting recent advances in the theory of prime numbers. Many of the great problems in mathematics, like Fermat's Last Theorem, have only been cracked once connections were made to other parts of the mathematical world. For 150 years many have been too frightened to tackle the Riemann Hypothesis. The prospect that we might finally have the tools to understand the primes has persuaded many more mathematicians and physicists to take up the challenge. The feeling is in the air that we might be one step closer to a solution. Dyson might be right that the opportunity was missed to discover relativity 40 years earlier, but who knows how long we might still have had to wait for the discovery of connections between primes and quantum physics had mathematicians not enjoyed a good chat over tea.

Marcus du Sautoy is professor of mathematics at the University of Oxford, and is the author of The Music of the Primes (HarperCollins).


TOPICS: Miscellaneous
KEYWORDS: 42; dyson; function; math; mathematics; numbers; numbertheory; physics; prime; quantum; riemann; zeta
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1 posted on 04/11/2006 3:09:02 PM PDT by LibWhacker
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To: PatrickHenry

prime ping?

;^)


2 posted on 04/11/2006 3:10:58 PM PDT by AntiGuv (The 1967 UN Outer Space Treaty is bad for America and bad for humanity - DUMP IT!)
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To: LibWhacker
"But the important role played by the number 42 has recently persuaded even the deepest skeptics that the subatomic world might hold the key to one of the greatest unsolved problems in mathematics."

Paging Douglas Addams

3 posted on 04/11/2006 3:11:53 PM PDT by muir_redwoods (Free Sirhan Sirhan, after all, the bastard who killed Mary Jo Kopechne is walking around free)
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To: LibWhacker
role played by the number 42 has recently persuaded

No Way!!! This is hilarious, the ultimate answer! God Bless Douglas Adams.
4 posted on 04/11/2006 3:15:26 PM PDT by tang-soo (Prophecy of the Seventy Weeks - Read Daniel Chapter 9)
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To: CarolinaGuitarman
Mathematics is not science?

Who told you that?

5 posted on 04/11/2006 3:15:48 PM PDT by muawiyah (-)
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To: LibWhacker

Humans are remarkable. We have the ability to view nature in a way that is very abstract and very often correct.

More than just random chance (IMHO).


6 posted on 04/11/2006 3:17:19 PM PDT by dhs12345
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To: LibWhacker

And of course, everyone knows the Primes went on to become the Temptations.


7 posted on 04/11/2006 3:17:59 PM PDT by stylin19a (I never put my foot in my mouth...I shoot that sucker off long before it gets anywhere near my mouth)
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To: muir_redwoods

I knew that something about the number 42 sounded "right" to me somehow. How did he know?

RIP DNA


8 posted on 04/11/2006 3:18:28 PM PDT by stands2reason
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To: LibWhacker
42
9 posted on 04/11/2006 3:18:33 PM PDT by Psycho_Bunny (The MSM is a hate group and we are the object of their disdain.)
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To: LibWhacker

There exists a simple, algebraic proof to fermat - I just don't have room for it in this post...


10 posted on 04/11/2006 3:19:14 PM PDT by patton (Once you steal a firetruck, there's really not much else you can do except go for a joyride.)
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To: muawiyah
I would agree with the statement that mathematics is not science because it doesn't follow the scientific method. You don't observe data, make a hypothesis, test it, and form a theory. In mathematics you just work from theorem to theorem based on axioms. While you can do something similar in the sciences, you are always constrained to the fact that if your theory doesn't match your data, the theory is wrong. In mathematics, the theorem *makes* the data.
11 posted on 04/11/2006 3:21:49 PM PDT by burzum (A single reprimand does more for a man of intelligence than a hundred lashes for a fool.--Prov 17:10)
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To: LibWhacker

I understand that quantum mechanics can also cook a mean omelet.


12 posted on 04/11/2006 3:22:19 PM PDT by stacytec (Nihilism, its whats for dinner)
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To: LibWhacker

bump


13 posted on 04/11/2006 3:23:06 PM PDT by tophat9000 (Illegals:”The Road to Serfdom” for the nation via Elites & Elect importing “better” serf’s)
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To: LibWhacker

I read it.
I reread it.
And I couldn't help thinking..."this article is about Prime Numbers, and they're talking about the number 42".
Which isn't a prime number.
I suppose it's the product of the first three primes.
Certainly the article could have added a little but more that air of hocus-pocus mystery (without, however, saying anything), by adding that detail, with a suitably raised editorial eyebrow of course.


14 posted on 04/11/2006 3:23:40 PM PDT by Vicomte13 (Et alors?)
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To: muawiyah

But it's the mathematical model that predicts the experimental results that gives the answer. It has to be. "Look at this picture" doesn't suffice in mathematics.

It's much like using a computer to get solution to some finite values. That gives you great instinct as to what the solution is, but you still need the proof.


15 posted on 04/11/2006 3:23:40 PM PDT by AmishDude (AmishDude, servant of the dark lord Xenu.)
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To: dhs12345
We have the ability to view nature in a way that is very abstract and very often correct.

If that were so, we wouldn't pay anybody to be a scientist. That we don't see nature very well at all makes scientists valuable.

16 posted on 04/11/2006 3:24:31 PM PDT by RightWhale (Off touch and out of base)
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To: stacytec

http://www.pithemovie.com/


17 posted on 04/11/2006 3:25:00 PM PDT by stacytec (Nihilism, its whats for dinner)
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To: patton

Ha! Inside joke for mathematicians!
Never heard of Seed magazine. Anyone know if it's generally worth reading?


18 posted on 04/11/2006 3:27:16 PM PDT by Buck W. (Avoid Evian Flu--Don't Drink Bottled Water.)
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To: Vicomte13

Oh hell, while I'm add it, I ought to add more hocus pocus and mystery.

42 has dramatic religious significance because it is the product of the first TRINITY of Primes!
And indeed, the first prime and second prime, multiplied together, produce 6, the number of man! So, the primes produce a man! But then the very next prime, following intimately and immediately thereafter, is the very next number, 7! The divine seven, the holy spirit following the product of the first two. Now, multiply the trinity of father, son and holy spirit primes (so to speak), and you get 42!!! The divine product of the number of man 6 (Which is the product itself of the first pair of primes, father and son), and 7, the holy spirit of primes!
Even more mysteriously, the sum of the digits of the product of the First Trinity of Primes is...SIX! Man re-emerges!
Holy bubbling bongwater, Batman!!!


19 posted on 04/11/2006 3:28:59 PM PDT by Vicomte13 (Et alors?)
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To: RightWhale
However, I believe that all possess the ability. Just that there are people who are more able than others. And yes, they get paid for it, albeit not that well sometimes. But their reward isn't money, is it?
20 posted on 04/11/2006 3:31:28 PM PDT by dhs12345
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To: Psycho_Bunny

21 posted on 04/11/2006 3:32:28 PM PDT by stacytec (Nihilism, its whats for dinner)
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To: dhs12345
I strongly disagree. Mathematics is a way for us to understand the world around us. Math is how our brain works. Nature doesn't like to behave the way we think.

The places where mathematics are most effective are those situations with the fewest assumptions -- particle physics, for example.

The more complex the system, the more it defies mathematical understanding and the "fuzzier" the subject.

22 posted on 04/11/2006 3:32:53 PM PDT by AmishDude (AmishDude, servant of the dark lord Xenu.)
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To: LibWhacker

Does this mean we don't need Deep Thought, or does it mean we do?


23 posted on 04/11/2006 3:34:12 PM PDT by js1138 (~()):~)>)
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To: stacytec
Great movie!

I'm looking forward to the sequel:

e

24 posted on 04/11/2006 3:35:31 PM PDT by who_would_fardels_bear
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To: LibWhacker

4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42... 4 8 15 16 23 42...


25 posted on 04/11/2006 3:35:34 PM PDT by Fudd (If at first you don't succeed, remove all evidence of your attempt.)
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To: burzum

A physicist, an engineer and a mathematician were all in a hotel
sleeping when a fire broke out in their respective rooms.

The physicist woke up, saw the fire in his room, ran over to his desk, pulled
out his CRC, and began working out all sorts of fluid dynamics
equations. After a couple minutes, he threw down his pencil, got
a graduated cylinder out of his suitcase, and measured out a
precise amount of water. He threw it on the fire, extinguishing
it, with not a drop wasted, and went back to sleep.

The engineer woke up, saw the fire, ran into the bathroom, turned
on the faucets full-blast, flooding out the entire hotel room,
which put out the fire, and went back to sleep.

The mathematician woke up, saw the fire, ran over to his desk,
began working through theorems, lemmas, hypotheses , you -name-it,
and after a few minutes, put down his pencil triumphantly and
exclaimed, "I have proven that I can put the fire out!"
He then went back to sleep.


26 posted on 04/11/2006 3:36:36 PM PDT by mwyounce
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To: Vicomte13
I don't want to burst your bubble or anything, but 5 is the next prime, not 7. (after 2 and 3)
27 posted on 04/11/2006 3:36:44 PM PDT by NathanR (Après moi, le deluge.)
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To: dhs12345
But their reward isn't money, is it?

Depends on ambition. There is some status for those at the PhD level. Status is important to some. Top slot at the Institute pays well. It is also an ego boost for those who know they are smarter than 99.99% of the population even if they don't hold a top position in industry or gov't.

28 posted on 04/11/2006 3:37:16 PM PDT by RightWhale (Off touch and out of base)
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To: mwyounce
Ah yes, he proved his existence theorem and then died.
29 posted on 04/11/2006 3:39:12 PM PDT by NathanR (Après moi, le deluge.)
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To: dhs12345
Humans are remarkable. We have the ability to view nature in a way that is very abstract and very often correct.

An intuitive speaks.

Be careful. FR is filled to the bursting with know-it-all positivists who will poke you with their pitchforks and burn you with their torches if you don't bow in abject humility before their peculiar icons.

30 posted on 04/11/2006 3:40:44 PM PDT by JCEccles
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To: LibWhacker

"What is so deep about this number 19?" Louis Farrakhan



31 posted on 04/11/2006 3:40:59 PM PDT by Right Wing Assault ("..this administration is planning a 'Right Wing Assault' on values and ideals.." - John Kerry)
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To: Mitchell

ping


32 posted on 04/11/2006 3:41:35 PM PDT by Allan (*-O)):~{>)
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To: Vicomte13

The third prime is 5.


33 posted on 04/11/2006 3:42:50 PM PDT by 2harddrive (...House a TOTAL Loss.....)
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To: mwyounce

Actually, after the engineer has finished, the mathematician wakes up, sees the fire going out and a smoldering ember burning the corner of his notes.

He immediately empties his whiskey bottle, pours out the contents all over the room, lights a match and goes back to bed...

...secure in the knowledge that he has reduced it to a previously-solved problem.


34 posted on 04/11/2006 3:43:12 PM PDT by AmishDude (AmishDude, servant of the dark lord Xenu.)
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To: patton

Have you tried the margin?


35 posted on 04/11/2006 3:45:13 PM PDT by DuncanWaring (The Lord uses the good ones; the bad ones use the Lord.)
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To: RightWhale
It is also an ego boost for those who know they are smarter than 99.99% of the population

It has just been in the past 20 years or so that a survey of professors' IQs at Cambridge University yielded a mean of 118.

Clever, yes. But hardly geniuses.

Academics in general are too impressed with themselves.

36 posted on 04/11/2006 3:45:34 PM PDT by JCEccles
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To: JCEccles

Anyone without a PhD is not considered a sentient being.


37 posted on 04/11/2006 3:48:42 PM PDT by RightWhale (Off touch and out of base)
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To: DuncanWaring

I, as a mathematician, am always operating on margin.


38 posted on 04/11/2006 3:49:06 PM PDT by patton (Once you steal a firetruck, there's really not much else you can do except go for a joyride.)
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To: LibWhacker

Okay, when did 42 become a prime number?


39 posted on 04/11/2006 3:49:12 PM PDT by Doohickey (Democrats are nothing without a constituency of victims.)
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To: AmishDude

So you are saying that mathematics wouldn't exist if we humans didn't?

Wouldn't the reality that we humans attempt to explain with math and science still exist?








40 posted on 04/11/2006 3:49:42 PM PDT by dhs12345
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To: muawiyah

"Mathematics is not science?

Who told you that?"

Mathematicians and scientists.


41 posted on 04/11/2006 3:50:02 PM PDT by CarolinaGuitarman ("There is grandeur in this view of life....")
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To: LibWhacker
It would mean that the primes behave rather like the random molecules of gas in a room: Although you might not know quite where each molecule is, you can be sure that there won't be a vacuum at one corner and a concentration of molecules at the other.

The frequency with which some part of a room will suddenly exhibit a substantial vacuum is - disturbingly - real and calculable. I believe it is many times the age of the universe for most examples, but it is something crazy to consider no less.

42 posted on 04/11/2006 3:52:41 PM PDT by SteveMcKing
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To: dhs12345
Wouldn't the reality that we humans attempt to explain with math and science still exist?

This question has been around a long time and the final answer is not in. Math and science majors should be encouraged to read some philosophy so they might have an idea what it is they are trying to do.

43 posted on 04/11/2006 3:53:44 PM PDT by RightWhale (Off touch and out of base)
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To: dhs12345
So you are saying that mathematics wouldn't exist if we humans didn't?

Well, I think so. But that's like a "tree falls in the forest"-type question.

Wouldn't the reality that we humans attempt to explain with math and science still exist?

I think you're conflating "science" with "nature". Science is a process. It's the application of the scientific method. It's the same thing as mathematics in that sense. If there's no one to perform the science, it will not occur. But nature will still be there.

44 posted on 04/11/2006 3:53:45 PM PDT by AmishDude (AmishDude, servant of the dark lord Xenu.)
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To: LibWhacker

Why do I have the feeling there must be some kind of Fatwa against this?


45 posted on 04/11/2006 3:54:45 PM PDT by colorado tanker (We need more "chicken-bleep Democrats" in the Senate!)
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To: patton

Doesn't that make you an economist?


46 posted on 04/11/2006 3:54:45 PM PDT by AmishDude (AmishDude, servant of the dark lord Xenu.)
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To: AmishDude

LOL, I'm sharing that joke and your addendum with my boss ( mathematician who wrote the Apollo trajectory software).


47 posted on 04/11/2006 3:55:04 PM PDT by stacytec (Nihilism, its whats for dinner)
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To: Vicomte13
I suppose it's the product of the first three primes.

I counter-suppose that the number "5" would disagree with you.

48 posted on 04/11/2006 3:56:12 PM PDT by krb (ad hominem arguments are for stupid people)
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To: Doohickey

42 isn't prime, but it is the "third moment of the Riemann zeta function," which in turn is important if you want to understand primes.


49 posted on 04/11/2006 3:56:15 PM PDT by LibWhacker
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To: NathanR

"I don't want to burst your bubble or anything, but 5 is the next prime, not 7. (after 2 and 3)"

No, no, no.
You're missing the point!
5 doesn't count for the Trinity of Primes, because it's just the sum of the first two primes, and when you sum the first prime and 5 (because everything has to be envisioned in a circle), you get 7! And that is why 42 is holy.
Don't you see now?


50 posted on 04/11/2006 3:56:23 PM PDT by Vicomte13 (Et alors?)
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