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Science

Prime Numbers Not So Random? 147

Posted by chrisd from the voltron-under-control dept.
Jeff Moriarty writes "Some physicists believe they might have caught a whiff of a pattern in the sequence of prime numbers. This would have a huge impact across mathematics, and to people who just really like primes... or like being Prime."
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Prime Numbers Not So Random? More

Prime Numbers Not So Random?

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  • anyone else getting the feeling... (Score:4, Interesting)

    by ddd2k ( 585046 ) on Tuesday March 25, 2003 @04:08AM (#5589593) Homepage
    the interval thing seemed like such a trivial observation. surely many others have easily noticed that. Its another "I think i discovered a pattern" claim, while still have no way to prove it.
    • In Mathematics, there's nothing that's "proven" that isn't explicitly defined as such. Notice how the Pythagorean Theorem is just that - a Theorem, not 'The Pythagorean Law'.

      The reason - it's impossible to prove anything on an infinite set of data that isn't defined in the parameters of the data set.

      A Theorem is a tested hypothesis, and these guys aren't even offering this. They're simply saying, "Look, we found an interesting pattern." As someone who's hopefully a future scientist, I'd say this is notewo
      • The reason - it's impossible to prove anything on an infinite set of data that isn't defined in the parameters of the data set.
        This is not physics, in math u can easily prove theories involving infite data sets. Hello? irrational numbers? infitie series? they were all logically proven. Its *NOT* noteworthy because anyone can come up with these observations, but it takes a genius to prove it.
        • There, you've done it.

          You've "easily" proven things by defining them as something. An irrational number is a number with no known, infinite, repeatable sequence? You've *defined* it that way, that doesn't mean you've ever *proven* a number irrational.

          People are still doing work on Pi to see if it's got repeatable, discernable patterns someplace. The application of Logic does not prove things, proof cannot be generated with interpolation/extrapolation. In the scientific community, proof is established by r
        • This is not physics, in math u can easily prove theories involving infite data sets. Hello?

          In English, you can easily use real grammar and real words. On slashdot, however, you are on your own.

      • While traditionally there has been a wide gap between physics and mathematics in terms of method, quite a few philosophers have argued that this gap has been closing. One big reason for it is the advent of the computer. Many of the most famous mathematical discoveries of the last decade have involved a *lot* of computer work. To such an extent that suggesting that anyone has actually gone through every step of the proof is likely wrong. Further studies in fields like chaos have tended to use computers l
      • The Pythagorean Theorem is a Theorem because it fits the definition for the Theorem, not because some stupid reason about infinity. A theorem is a logical assertion which can be proved using (=it is implied) by the axioms (statements which are given to be true).

      • In Mathematics, there's nothing that's "proven" that isn't explicitly defined as such. Notice how the Pythagorean Theorem is just that - a Theorem, not 'The Pythagorean Law'.

        I don't see any difference between a "law" and a "theorem".

        Anyway, a theorem is a formula that can be proven true.

        Formulas that aren't theorems are, well, just formulas.

        All the math we know consists of theorems, things that have been proven true. There are also some so-called "conjectures" - that means "we think this is a theor

        • Unfortunately, not everything in math can be a theorem.
          We know since Gödel that some truthes are just "truthes", like mere accidents, which means, they cannot be proven with our set of axioms.
          For this precise reason, the mathematics universe is starting to look a lot more like the physicis universe, in that laws might be an option to consider.
          Say we discover an appearant pattern in prime numbers distribution. Maybe this pattern, experimentaly found has no way to be proven.
          The real bad news is, if i

          • in that laws might be an option to consider.

            What are these "laws" you are talking about? Things aren't special because they have "law" in their name (for instance, in Dutch it's "Law of Pythagoras" not "Pythagoras' Theorem" - doesn't mean anything different).

            But ok, I cede that there are conjectures that cannot be proven and still be true. But many examples of long-lived conjectures (the "four colour theorem", "fermat's last theorem") were eventually proven. Just from looking at Gödel's constructe

        • I don't see any difference between a "law" and a "theorem

          The reason the term law is not used is because a law is something that has to always hold true.

          On the other hand, a Theorem is something that is based on a set of axioms. It may change, within the limitations of the axioms or even independent of them.

          From a Physicist's perspective, both Newtonian mechanics and Relativistic mechanics hold true, but you do not consider relativistic mechanics for your day-to-day problems in Physics. Which is why a la
      • Notice how the Pythagorean Theorem...

        Like this? [pnl.gov].

        Took about 30 seconds with google, and that's because I misspelled Pythagorean. Good thread, however.

      • it's impossible to prove anything on an infinite set of data that isn't defined in the parameters of the data set.

        Of course it's possible. Between high school, college, and grad school I'm sure I proved hundreds of propositions about infinite sets.

        A Theorem is a tested hypothesis, and these guys aren't even offering this.

        No.

        A theorem is a proven mathematical statement. E.g., the Pythagorean theorem, or the fundamental theorem of integral calculus.

        A theory is (in a scientific context) a tested

    • I agree. This doesn't sound like anything different from all the patterns (real or not) discovered by this writer playing with his TI-30 while bored in math class.

      Of course, the article was sparse on details, but it seems they are taking a physical sciences approach to mathematics... make a hypothesis and experiment to see if it is correct. In math, this is useless because as a system of logic, mathematical proofs can be either proven or disproven 100% (Godel exceptions notwithstanding). That never happ
    • Well, it's a well proven fact in mathematics that given enough random points on a graph, patterns have to emerge.

      I don't remember the exact name of the theorem, but Erdös, and his contemporaries were main figures in the development of it.

      Given a billion prime numbers, you have essentially a billion points on a graph, naturally meaningless patterns are going to emerge. It doesn't really tell you anything though.

  • Encryption? (Score:3, Informative)

    by asdfx ( 446164 ) on Tuesday March 25, 2003 @04:15AM (#5589603) Homepage
    I wonder if this theory could be used to produce code that could be useful for encryption based on prime numbers, such as RSA's work. Would it make it easier to produce reliable prime numbers much larger than 1024 or even 2048 bit? Further, I wonder if this could be used to drastically reduce the time required to brute force an RSA encrypted message. Could the encryption of files that were encrypted with 128 bit technology be rendered all but useless?

    • Re:Encryption? (Score:2, Insightful)

      by itsme1234 ( 199680 )
      Uh, I fail to see why this was moderated as interesting. "128 bit technology" ? I assume you are talking about symmetrical alg., like IDEA, CAST, and many others. These are not even remotely related to prime numbers (some of them are, but not very close). And it's already simple enough to generate big prime numbers.
      Next step is to ask: "will my Diesel car become obsolete because of this theory" ?
      • He was talking about RSA. And yes, if someone figures out a way to use these patterns to calculate large prime numbers more quickly, this could definitely have huge consequences on asymmetrical encryption like RSA. Depending on how much faster it makes calculation of large primes, it could either require much larger RSA keys (say 40kbit instead of 4kbit) or even make RSA and other prime-number-based encryption schemes inherently insecure.

        Daniel
        • RSA is "strong" not because we cannot generate prime numbers (or test if one given number is prime). In fact we can do both things VERY fast.

          RSA is "strong" because we cannot solve fast simple ecuations like x*y=A (where A is BIG, x,y integers). And bruteforce is NOT the fastest method available to factor integers. If it were, yes, it would help to have a faster algorithm to generate/test primes. But it's not.
          • Wouldn't it be pretty fast if you start with A, put z = sqrt(A), then search around z already knowing all the prime numbers from 0 to z? If the method accelerates things enough, it's equivalent to knowing all the numbers from 0 to z. I would have thought that going through a few zillion numbers and trying to divide to see if we get an integer shouldn't take that long - at least it would take many orders of magnitude less time than trying ALL the integers in there.

            Oh, and from what I've seen, it does take
  • We have always maintained that it is not random. In fact, our random number generator consistently generate numbers that are subsequently found to be NON-PRIME.

    In our extensive (yet to be published) research, we have discovered that all PRIME NUMBERS are not just not random, but are found to have the property of NOT HAVING ANY DIVISORS APART FROM ITSELF AND 1. I've yet to verify with finding but it appears to be true with a correlation of 1.0 for all cases our research team have considered.

    • figure & ground (Score:4, Interesting)

      by obtuse ( 79208 ) on Tuesday March 25, 2003 @04:46AM (#5589692) Journal
      Yeah, I remember being excited when I saw a graph of primes that were dots in a field of blank composites. There were lines & patterns all over the place. Wow!

      Then I realized that the composite numbers will each make a pattern in any graph. By their nature they repeat.

      What I was looking at was the space in between the patterns created by the composites. For example, all primes are odd. There's a set of straight lines on any graph. Well, it's more enlightening to say that none are even, becasue then they'd be divisible by two. Each new set of composites creates another pattern that makes a hole in possible primes.

      • Re:figure & ground (Score:1)

        by Anonymous Coward
        For example, all primes are odd.
        Uh, hello? 2?
        • For example, all primes are odd.

          Uh, hello? 2?

          That's just a measuring error :-)

        • 2 is the oddest prime number of all, 'cause it doesn't obey the rule all the others do of not being divisible by 2.

          Yeah, I know. Terrible pun. So, if anybody ever figures out how to define division by zero, will this screw up the definition of a prime number?

          • Uh, it *is* divisible by two. 2/2=1
            • And because it is divisible by 2, it does not follow the rule followed by all other prime numbers of *not* being divisible by 2, hence making it an "odd" prime number in spite of being "even". You read my previous post too quickly, apparently.
          • Excuse my extreme naivete, but if division is measuring how many times you can take one number from another, wouldn't dividing anything by zero give you infiniti?

            IE: 4 / 2 = 2 because you can remove 2 twice from 4 before you get a number less than 2 (0).

            So then, since removing 0 from anything has no effect, you can do it an infinite number of times with no change, right? (Kinda like a geek asking a babe for a date... heh)
            • I like the other guy's positive and negative infinity explanation but the way I explain it to myself is that zero isn't just the abscence of something, it's actually nothing, and anything that's something doesn't have any nothings in it to remove, although I admit that even that explanation falls apart when you consider 0/0. (It seems to me that zero should go into zero exactly one times, but I'm sure there must be one of those chain of equations that starts there and ends with 2=3 or something like that.)
    • Please mod the parent up, but NOT as funny. And move along, there is nothing else to see here.
    • The funniest part of this post is that it got moderated "insightful". Hahaha.
  • I don't really know anything about number theory, but I I get a little suspicious when anyone announces a discovery in a field unrelated to their area of expertise. Utah chemists did this in 89 or 90 with Cold Fusion, and it was quickly shown to be bad science by physicists.

    Can anyone out their study number theory give us a heads up if they may be on to something, or this is simply just crazy?
    • This isn't some obscure result that requires sensitive equipment and millions of dollars worth of lab equipment. It's a simple proposition about numbers. Read the article. Do it yourself. It's not hard.
    • As a number theory graduate student, this looks suspicious. This isn't as bad as last summer, when some string theorists claimed a junk [slashdot.org]
      proof of the Riemann Hypothesis, but it's close.

      Prime numbers are very hard to tackle. Part of the difficulty in this style of problem, as another post points out, is that they are defined multiplicatively, and yet we here care about additive properties (differences in this case).

      I have a few concerns with this paper:

      1. They look at a really small number of primes (onl
      • Another point:

        One page 14 (figure 5) they discover the following fact: the difference between the i'th and (i+1)st primes is about log(i).

        That is exactly the prime number theorem I mentioned above, conjectured by Gauss around 1800 and proved in 1896 by Hadamard and de la Valee Pussin.

        Writing a paper on the distribution of primes and not referring to that is like writing a paper mentioning a discovery that planets move in elliptical orbits, while being ignorant of Kepler's laws or Newton's explanation of

  • Here's the rub (Score:5, Interesting)

    by Anonymous Coward on Tuesday March 25, 2003 @05:20AM (#5589772)
    Here's the problem with finding patterns in Primes: It has to do with the way most things in number theory are formulated. Prime numbers are figured out by a process of non-definition and NOT by some form of additive process. An example or two might make that statement a bit clearer:

    If I needed, for example, to find a rule that returns only even numbers, my problem is simplicity itself, I have no need to test a given number to determine whether or not it is even, I can force it to be even by applying any number of simple (or complex) formulas that work within the system.

    If someone gives me number X, I have no need to know what X is, all I have to do is multiply X by 2 and (after a little inductive reasoning), I have guaranteed that I now have an even number.

    Prime numbers are NOT found that way. An even number is determined to have the property 'evenness' from within the number system itself, namely multiplication by 2. It is a simple additive process to include other even numbers into a given set. A prime number on the other hand, forgive the inexactness, can be considered to have the inherent property 'whatever property that created me that is unique to me'.

    IOW, each prime number is unalterably unique and furthermore it is unique in a way which is unique to EACH AND EVERY prime number, all by itself. No other prime number has the same property that makes any other prime number unique.

    EXAMPLES (bad, I know, but the best I could do at 0430):
    the number 7 (a prime) has the unique property (among other properties, like 'oddness') that it has the unique divisors 7 and 1, a property that it shares with no other numbers.

    the number 17 (a prime) has the unique property (among other properties, like 'oddness') that it has the unique divisors 17 and 1, a property that it shares with no other numbers.

    the number 21 (not a prime) has the property (among other properties, like 'oddness') that it has the divisors (7 and 3) AND (21 and 1). Only primes get to leave out that AND part.

    The prime numbers are the GAPS within the number-system (and in a rather pathological side note - they are also the glue that holds the system together). The definition of a prime number is, put simplistically: ANY number X that is NOT composite.

    Saying you have found a pattern in the prime numbers is tantamount to saying that you have a rule that can create prime numbers W/O checking to see if it's true or not. Put another way, it is exactly the same as saying:

    "I have a formula P(x) that can always churn out primes, give me a number, any number and after the application of my formula, I can guarantee that it will be a prime number."

    If you could do that, I have a whole bunch of NP complete problems for you to work on (and a bone to pick with a certain Mr. Godel).

    Any pattern w/in the set of prime numbers would be a formula with an infinite number of rules (an individual rule for each individual prime number, AT LEAST), and anything with an infinite number of rules can be considered completely, totally and irrevocably RANDOM.

    Some late night ramblings from a guy who's too tired and lazy to log on.

    • Re:Here's the rub (Score:3, Informative)

      by Anonymous Coward
      They don't claim that they have a rule that can create prime numbers, they just claim that prime numbers might not be completely random.

      Just like if you have a large prime p, p+210 is 4.375 times more likely to be a prime than a random integer around p. Not a rule, but a hint that primes aren't so random.

    • the number 7 (a prime) has the unique property (among other properties, like 'oddness') that it has the unique divisors 7 and 1, a property that it shares with no other numbers.

      the number 17 (a prime) has the unique property (among other properties, like 'oddness') that it has the unique divisors 17 and 1, a property that it shares with no other numbers.

      The number 15 (not a prime) has the unique property (among other properties, like 'oddness') that it has the unique divisors 3 and 5, a property tha

    • Not multiples of (pick a number other than 1 & the prime.) They're defined by the patterns they don't fit. That looks like an irregular or near fit to a pattern.

      I said all primes are odd in an earlier post. Sorry, all primes but the number two are odd.

      I hacked up a perl script to demonstrate what these guys were describing. I don't want to drop it in here, because it's a shameful late night hack, but it's in my journal. It generates primes, increments, intervals, and a running total of the intervals,

      • results of your script:

        ./primer.pl Number found where operator expected at ./primer.pl line 7, near "] 1" (Missing operator before 1?) Number found where operator expected at ./primer.pl line 28, near ") 8193" (Missing operator before 8193?) syntax error at ./primer.pl line 7, near "] 1" syntax error at ./primer.pl line 28, near ") 8193" Execution of ./primer.pl aborted due to compilation errors.

        Sorry :-)

    • I've got one! (Score:3, Funny)

      by Andy_R ( 114137 )
      "I have a formula P(x) that can always churn out primes, give me a number, any number and after the application of my formula, I can guarantee that it will be a prime number."

      If you could do that, I have a whole bunch of NP complete problems for you to work on (and a bone to pick with a certain Mr. Godel).

      x-x+7 gives a prime number for every value of x ;-)

    • "I have a formula P(x) that can always churn out primes, give me a number, any number and after the application of my formula, I can guarantee that it will be a prime number."

      That's trivial. P(x+1)=1+PI{P(i) for i = 0 to x}, P(0) = 1 or 2, depending on whether you want to list 1 as a prime number. That's been know since antiquity.

      What would blow open mathematics would be a non-trivial function to determine all prime numbers, in order, with at most a finite number of known omissions.

    • > "I have a formula P(x) that can always churn out primes, give me a number, any number and after the application of my formula, I can guarantee that it will be a prime number."

      A non trivial formula, you mean. Otherwise the following applies:

      P(x) = 7(x/x)

  • "3 is prime, 5 is prime, 7 is prime, 9 is, um, experimental error, 11 is prime, 13 is prime--looks good."
  • Hey, I think I found a pattern to the prime numbers!!!1!! While I admit I haven't had a chance to try them all, it looks like primes greater than 3 are of the form 6n-1 or 6n+1, where n is an integer.

    5=6*1-1, 7=6*1+1, 11=6*2-1, 13=6*2+1, 17=6*3-1, 19=6*3+1, ..., 3141592799=6*523598800-1, 3141592801=6*523598800+1, ...

    Pretty cool, huh? So where's my Field's Medal? Or at least I should get published in Nature for this!

    • 3141592799 / 3 = 1047197600

      So not a prime.

      Daniel
      • You might try setting your calculator so that it doesn't round to integer. 3141592799 is prime. 31415927299 / 3 = 1047197599 + 2/3.

        P.S. Another thing worthy of a Nature article... an integer is evenly divisible by 3 if the sum of its digits is evenly divisible by 3. 3+1+4+1+5+9+2+7+9+9=50. 5+0=5. 5 is not evenly divisible by 3. Therefore neither is 31415927299.

        P.P.S. 1047197600 has two zeros at the end. If you multiply it by any integer, the product will have at least two zeros at the end. Therefore a tr

    • Re:I found a pattern! (Score:5, Insightful)

      by itsme1234 ( 199680 ) on Tuesday March 25, 2003 @09:07AM (#5590290)
      Of course they are prime ! ANY number is either:

      6n (not prime of course)
      6n+1
      6n+2 (not prime of course)
      6n+3 (not prime of course)
      6n+4 (not prime of course)
      6n+5

      And 6n+5 is the same as 6(n+1)-1 so indeed you are right. You deserve a price for finding a 6th grade theorem.
      • Of course they are prime ! ANY number is either:

        6n (not prime of course)
        6n+1
        6n+2 (not prime of course)
        6n+3 (not prime of course)
        6n+4 (not prime of course)
        6n+5

        Do you mean for all integers n = 1, 2, 3, ... or do you start at zero? Either way, what about the number 3? If you start at n=0, then 6n+3 = 3, but you claim that 6n+3 is "not prime of course," so your claims need to be checked more carefully.

        However, I agree with the overall point of your post.

      • You deserve a price for finding a 6th grade theorem.

        Yes! Fields Medal! Now!! Before I get too old for it...

    • ALL primes will have to be of the form:
      2n+1 (not divisible by 2)
      3n+1, 3n+2 (not divisible by 3)
      5n+1, 5n+2, 5n+3, 5n+4 (not divisible by 5)
      etc for all Pn+d , where P is prime, n is an integer, and d is an integer 1<=d<P

      So, your theorem is correct, as all primes will have to fit the form ((2n+1) AND ((3n+1) OR (3n+2))), which can be written as ((6n+1) OR (6n-1)), but unfortunately, this does not help much.

      It would be much more helpful to find an equation stating that all numbers of the single form (..
    • Ding! Correct!
      6(9) - 1 = 53
      6(56) + 1 = 337

      Bzzt. Incorrect!
      6(9) + 1 = 55
      6(56) - 1 = 335

      What you need is a GUT (grand unified theory). It's like trying to combine quantum mechanics with general relativity. Good luck. :)
    • That doesn't work for 5^2 (6*4+1=25). Let's extend that by saying primes greater than 5 are in the form 30n-13, 30n-11, 30n-7, 30n-1, 30n+1, 30n+7, 30n+11, or 30n+13, where n is an integer. Surely we've now got a working method!

      But that doesn't work for 7^2 (2*30-11), so we'll modify that to : Primes greater than 7 are in the form 210(=2*3*5*7) plus or minus one or (any of the prime numbers less than 105 and greater than 7).

      But that doesn't work for 11^2(210-89), so...

      Repeat this process for all the pr

  • Anyone can test this theory out. (Score:3, Interesting)

    by pauldy ( 100083 ) on Tuesday March 25, 2003 @10:08AM (#5590540) Homepage
    Take the first 1000 primes from the site listed. Put them in your favorite spreadsheet. Then use the formula they give to find out they are mostly full of it. For they first few it looks like a pattern is forming then it looks like nothing but noise when plotted. I can't believe no one even tryed this before they actually published this article.

    • - I can't believe no one even tryed this before they actually published this article.

      Well, they did. Thats what all the above gags are about, that these physicists are unaware of prior, basic "Fun with Primes!" work. Nature too, evidently.

      • Was this specific result already found?

        I only read the abstract, but it seems they were only looking at the 'increments' between the gaps in prime numbers.

        The gaps between prime numbers have been well studied, but perhaps no one has bothered to look at the increments.

        Still, I agree that this is does not look all that surprising since the distribution of primes is well-studied, but they may have looked at some wrinkle that people had not looked at before.

        Also, the Nature write-up was particularly clueles
  • wow, another random non-mathematician finding isolated patterns in a mathematically complex sequence of numbers...

    the patterns they describe are likely nothing more than side effects that can be produced using a number sieve. that seems to be what most of the "prime formulas" that people come up with can be reduced to.

  • Parallels with Carl Sagan's "Contact" (Score:1, Interesting)

    by Anonymous Coward
    This sounds spookily like the ending in Contact (only in the book, not the film) where researchers find a message buried in the seemingly random digits of Pi. The implication was that the builders of the universe had left behind their signature.

    Perhaps these guys should map out their sequences of prime number differences to see if it generates a picture ?

  • Skip right to the paper. (Score:3, Informative)

    by molo ( 94384 ) on Tuesday March 25, 2003 @12:00PM (#5591387) Journal
    Want the details? Ignore the watered-down article and skip right to the research paper.. all greek to me, but has some interesting plots:

    Information Entropy and Correlations in Prime Numbers -- Abstract [lanl.gov]

    Information Entropy and Correlations in Prime Numbers [PDF] [lanl.gov]

    Information Entropy and Correlations in Prime Numbers [Postscript] [lanl.gov]

    -molo
  • So here is 5 minutes of mathemetica output to look at. dlist is the list of differences between sequential primes. ilist is the increment between subsequent differences. Have fun.

    ilist = dlist = {};
    Do[dlist = Append[dlist, Prime[n + 1] - Prime[n]], {n, 1000}];
    Do[ilist = Append[ilist, dlist[[n + 1]] - dlist[[n]]], {n, 999} ];

    dlist
    {1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2,

  • Don't forget the Prime Spiral [wolfram.com].

    This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam's slightly more formal prose, it "appears to

  • Hold on. You have to slow down. You're losing it. You have to take a breath. Listen to yourself. You're connecting a computer bug I had with a computer bug you might have had and some religious hogwash. You want to find the number 216 in the world, you will be able to find it everywhere. 216 steps from a mere street corner to your front door. 216 seconds you spend riding on the elevator. When your mind becomes obsessed with anything, you will filter everything else out and find that thing everywhere.

    2*2*

  • They're intimately tied to their position along the integer continuum. It's just that the complexity of determining primality (the information content, in fact) increases with the position.

    Randomness is not actually entropy.

    --Dan
    • Right, Primes have intrinsic information content, so we shouldn't be surprised someone could measure their entropy if they tried. These physicists used their own data-analytic tools to measure this in an empirical way. Whether this provides a new insight to the number theorists or not is yet to be seen, but applying new tools to problems with which mathematicians have been "stuck" has sometimes provided a needed boost.

      The pattern they've found is a logarithmic distribution, it seems, according to their abs

  • Now, saying "all primes but 2 are odd", is just the same as saying "all primes except 2 do not divide evenly by 2".

    FWIW, I can offer the following additional observation: All primes except 2 and 5 must end with 1, 3, 7 or 9, and these must be matching one of:

    30n+7 30n+11 30n+13 30n+17 30n+19 30n+23 30n+29 30n+31

    for all n>=0

    I guess similar arguments may be made for including further factors 7 (210n+7 etc) and 11 (2310n+7 and so on) but I suspect this gets too unwieldy too soon to be very useful.

    F

  • "It was mentioned on CNN that the new prime number discovered recently is four times bigger than the previous record." John Blasik

    "You know what seems odd to me? Numbers that aren't divisible by two." Michael Wolf.

    "I don't get even, I get odder."

    Well, the problem "How to prove that all odd numbers are prime" has different solutions whether you are a:

    Mathematician: 1 is prime, 3 is prime, 5 is prime, 7 is prime, and by induction we have that all the odd integers are prime.

    Physicist: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is an experimental error...

    Engineer: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime...

    Chemist: 1 prime, 3 prime, 5 prime... hey, let's publish!

    Modern physicist using renormalization: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is ... 9/3 is prime, 11 is prime, 13 is prime, 15 is ... 15/3 is prime, 17 is prime, 19 is prime, 21 is ... 21/3 is prime...

    Quantum Physicist: All numbers are equally prime and non-prime until observed.

    Professor: 1 is prime, 3 is prime, 5 is prime, 7 is prime, and the rest are left as an exercise for the student.

    Confused Undergraduate: Let p be any prime number larger than 2. Then p is not divisible by 2, so p is odd. QED

    Measure nontheorist: There are exactly as many odd numbers as primes (Euclid, Cantor), and exactly one even prime (namely 2), so there must be exactly one odd nonprime (namely 1).

    Cosmologist: 1 is prime, yes it is true....

    Computer Scientist: 1 is prime, 10 is prime, 11 is prime, 101 is prime...

    Programmer: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 will be fixed in the next release, ...

    C programmer: 01 is prime, 03 is prime, 05 is prime, 07 is prime, 09 is really 011 which everyone knows is prime, ...

    BASIC programmer: What's a prime?

    COBOL programmer: What's an odd number?

    Windows programmer: 1 is prime. Wait...

    Mac programmer: Now why would anyone want to know about that? That's not user friendly. You don't worry about it, we'll take care of it for you.

    Bill Gates: 1. No one will ever need any more than 1.

    ZX-81 Computer Programmer: 1 is prime, 3 is prime, Out of Memory.

    Pentium owner: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 8.9999978 is prime...

    GNU programmer: % prime
    usage: prime [-nV] [--quiet] [--silent] [--version] [-e script] --catenate --concatenate | c --create | d --diff --compare | r --append | t --list | u --update | x -extract --get [ --atime-preserve ] [ -b, --block-size N ] [ -B, --read-full-blocks ] [ -C, --directory DIR ] [--checkpoint ] [ -f, --file [HOSTNAME:]F ] [ --force-local ] [ -F, --info-script F --new-volume-script F ] [-G, --incremental ] [ -g, --listed-incremental F ] [ -h, --dereference ] [ -i, --ignore-zeros ] [ --ignore-failed-read ] [ -k, --keep-old-files ] [ -K, --starting-file F ] [ -l, --one-file-system ] [ -L, --tape-length N ] [ -m, --modification-time ] [ -M, --multi-volume ] [ -N, --after-date DATE, --newer DATE ] [ -o, --old-archive, --portability ] [ -O, --to-stdout ] [ -p, --same-permissions, --preserve-permissions ] [ -P, --absolute-paths ] [ --preserve ] [ -R, --record-number ] [ [-f script-file] [--expression=script] [--file=script-file] [file...]
    prime: you must specify exactly one of the r, c, t, x, or d options
    For more information, type "prime --help''

    Unix programmer: 1 is prime, 3 is prime, 5 is prime, 7 is prime, ...
    Segmentation fault, Core dumped.

    Computer programmer: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime, 9 is prime, 9 is prime, 9 is ...

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