Prime Number Symmetry


Most texts on Prime Numbers (and even a professor of Maths at Oxford University?) don’t seem to explain the patterns and order in the primes and instead prefer to use the words “random” and “mysterious”. There is no doubt their distribution has been difficult to explain but just because something is “complex” does not make it random or unexplainble . I hope you find the diagrams below useful in understanding prime distribution and for predicting where primes can occur and where they definitely won’t. Comments welcome as I am learning all the time.

JH

Lines of Symmetry

Symmetry is not given much attention when considering prime number distribution. Hardly surprising as prime numbers don’t occur symmetrically! However if one looks at the positions that prime numbers occur it is clear that they occur only in certain locations, and don’t occur in other locations. In addition, the locations where primes do occur can be mapped using lines of symmetry.

The first couple of lines of “symmetry” are well known. The first reflects the fact that most (not 2 or 3) prime numbers can be expressed in the form 6n+1 and 6n-1, but with the disappointing aside that not all such numbers are prime. Within each strand of primes they are distributed at intervals of 3# = 6. As with further lines of symmetry, the smallest prime numbers don’t fit in, but in a predictable way – the patterns are stepped, becoming more complex the larger the integer. Thus the symmetrical patterns do not all start at 2, but at increasing primes. This is fundamental to understanding how primes are distributed.

Thus, putting 2 and 3 to one side, the prime numbers can be considered to be either left handed with a formula of 5 + n.3# (5,11,17,23…) or right handed with a formula of 7 + n.3# (7,13,19,31….). In theory at least the line of symmetry starts at 3 (or 3#/2).

The next line of symmetry starts at 5#/2 = 15 AND the pattern it explains starts at a larger prime, 11. The primes occur at intervals of multiples of 5#=30 from 8 bases:

 
 

with ALL prime numbers distributed at equal intervals about the line 15-45-75-105…. provided that smaller primes are discarded first. This diagram reflects a number of features of the primes which are obvious in retrospect:

i) there are only three types of pair of prime numbers 1:3, 7:9 and 9:1.

ii) the 1:3 pairs match the locations of the 7:9 pairs about the line of symmetry; the 1:9 pairs are equidistant from the line of symmetry

iii) there are two lines of primes that never form pairs (the +8 and -8 lines),

iv) there are zones of 6 integers on either side of the line of symmetry in which primes never occur (i.e. between the +8 and +14 lines) between primes ending in _1 and _7 on the one hand and those ending between _3 and _9.

In consequence it can be conjectured that there are an infinite set of similar lines of symmetry in the prime numbers each starting at n#/2; thus

105…315

1155…3465

15015…45045

etc

The lines of symmetry are not immediately obvious if one only has the set of prime numbers to start from. It is necessary to start from the Sieve of Eratosthenes in order to obtain the necessary groupings (to be described later).

The third line of symmetry starts at 105; primes occur at intervals of 7#=210. As shown it is not quite symmetrical and two lines from the bottom – starting 211 and 221 – have to be brought up to the top, but staggered, to provide the complete pattern. Some of these bases aren’t prime …..121, 143, 169, 187, 209, 221…but are predictable. Notice again the pairing of prime pairs about the line of symmetry, and the absence of primes in some zones of length 6, and some areas with no pairs:

Jon Heuch

Bringing the last two lines up to the top gives the complete picture:

[to be repaired and continued after someone hacked in here]

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20 Responses to Prime Number Symmetry

  1. Mr WordPress says:

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  2. Hugh says:

    My suggested approach to these symmetries is to look within the 6n+1 line and the 6n-1 lines in isolation. Then you see an even clearer symmetry. Within the 6n-1 line 5s and 7s are symmetrical from 35 to 245. Within the 6n+1 line they are symmetrical from 175 to 385. And the two lines can be considered as one if you imagine a mirror at zero “reflecting from one to the other”:

    11, 5, 1, 7, 13, 19 etc.

    Viewed like this the Sieve of E forms a lovely clear fractal pattern, with one in five numbers removed by 5, one in seven by 7 and so on.

    You might also want to play with a matrix/spreadsheet with the 6n-1 numbers on one axis and the 6n+1 on the other axis. Prime numbers can only appear in these axes. The products of the two numbers multiplied together give you other 6n+1 and 6n-1 numbers in fractal patterns, which are very revealing. The numbers in this matrix are also all the difference between two square numbers. Prime conjectures such as the Goldbach and Polignac conjectures get a nice visual representation in such a matrix as the products of the series of pairs run along the diagonals in arithmetic progressions.

  3. Søren Nielsen says:

    Read your site. Agree partly with you, special about the silly use of the word mysterious. If you send an email to srentospace@hotmail.com, I will return my ebook about the ulam spiral and other primepatterns.

  4. Gary Croft says:

    The Croft Spiral Sieve, a deterministic algorithm populated by the set of all natural numbers not divisible by 2, 3 and 5 bounded in an infinitely expanding 30-sectioned spiral, evenly distributes all prime numbers >5 along 8 radii and demonstrates how primes are arrayed within a beautiful geometry involving perfect mod 30 symmetry. Essentially, this sieve defragments the Ulam Spiral.

  5. James says:

    -”The Croft Spiral Sieve, a deterministic algorithm populated by the set of all natural numbers not divisible by 2, 3 and 5 bounded in an infinitely expanding 30-sectioned spiral, evenly distributes all prime numbers >5 along 8 radii and demonstrates how primes are arrayed within a beautiful geometry involving perfect mod 30 symmetry.”

    You have got to be kidding me. You want to know why there is “perfect mod 30 symmetry”? You took out 30′s prime divisors. And it’s hilarious that you named something completely meaningless after yourself in hopes that you would become famous once your brilliancy becomes a household name. I now name the set of all natural numbers except 27 and 108 as the Jameset. Once you realize the significance of this, you will be forced to say my first name every so often, which gives me pleasure.

    You cranks are hilarious. Want to know why there is a “prime spiral” (as read on Terence Tao’s blog, posted by one of you nuts in the comments) with respect to even and odds and primes? You are projecting your insanity onto paper. Differences between points mean that you could draw a damn zebra between them, and having delusions of grandeur would result in you taking them seriously. Something along the lines of: “Today I found something truly magnificent. I attempted to draw a zebra around the number pi, and upon doing so I realized that this zebra represents a unique collection of points. Indeed, each point in the zebra outline was real, and thus I have come to the mathematical oddity that zebras love pi.”

    Basically, stop doing math. You all have never made a single contribution (none of you, ever) and you never will. It gets annoying seeing you all pervert perfectly good math (i.e.: using Tao’s blog as a springboard for your lunacy).

    • Hi James. Had a bad day at the office? I don’t know who or what the Croft Spiral Sieve is – someone posted on this(my) website.

      You posted the question “Want to know why there is a “prime spiral” (as read on Terence Tao’s blog, posted by one of you nuts in the comments) with respect to even and odds and primes?”. Well yes I do! And it wasn’t me that posted the link between my website and Terence Tao – an unknown (to me) third party. And my website is not for fame or fortune but because of a simple interest in the matter. My posting is almost anonymous so you would have to be quite clever to list fame as a motivation!

      So, whilst the reasons for patterns and “pseudo-symmetry” about n#/2 are pretty obvious, they clearly aren’t very obvious at all. Why would a Professor of Mathematics at Oxford University use the word “random” to describe prime number distribution, when clearly they are far from random. And I can’t find any reference on Google (not a single entry) to p+np# as a formula for prime numbers – clearly it is NOT THE formula for prime numbers but it seems to provide some light in a world of darkness.

      This website may not make much of a contribution to “math” but I do hope that it assists in understanding and education, as mathematicians have been pretty awful (=failed) at communication.

      • Gary Croft says:

        Not everyone shares James’ opinion of the Croft Spiral Sieve given that there’s an MIT licensed Python module dubbed “pyprimes” designed to run and compare prime number sieving algorithms, including the Croft Spiral Sieve, which the programmer has rated “effective and fast.” http://pypi.python.org/pypi/pyprimes/0.1.1a#downloads .
        There’s much talk about the “decline of civility,” and James’ comments are a case in point. What I’d appreciate is a serious point-by-point critique from James, rather than a venomous, name-calling rejection (focusing on the person rather than the math). And telling amateurs that they should abandon their fascination with symmetry and patterns, which–after all–is the very definition of doing math, is ridiculous if not offensive.
        There was a time (late 19th & early 20th century) when collegiality between professional and amateur mathematicians was the norm and conjecture and experimentation were encouraged. Modern mathematics, on the other hand, has become a well-guarded fortress impenetrable to amateurs. I speak from personal experience. Long before posting my sieve on-line I made numerous attempts to communicate my ideas to mathematicians around the world. The number of responses, in mathematical terms = 0. And that is why, in desperation, amateurs like me go “open source.” In today’s mathematical universe of privileged knowledge protected by snarling gatekeepers {where proofs are holier-than-thou}, Ramanujan would be dismissed as a crank.
        There’s a middle ground between chaos and proofs called conjecture. Conjecture is the witch’s brew from which magic can and does emerge. What drives me is the thrill of discovery. It doesn’t matter one whit to me whether my “discoveries” are novel or naive: I’m still left with the thrill of finding what I consider to be beautiful patterns that when looked at holistically tell a compelling story that DOES help demystify that which has been declared “out of bounds” by the James’s of the world.
        The Croft Spiral Sieve is effectively an in-depth analysis of the set of all natural numbers not divisible by 2, 3 and 5 when arrayed in 8 columns (or 8 spiral radii) using nothing but elementary arithmetic accessible to any high school educated person. It attempts to show how the prime number sequence is birthed in a number chamber that at its core possesses breathtakingly beautiful progressions and symmetries (at least from my apparently ignorant perspective). For those of us who chose professions other than mathematics (in my case, former Sr. Director, Global Procurement and Business Operations at Microsoft) your sieve and mine DO demonstrate algorithmically how the prime number sequence is formed. The last thing I care about is fame and glory … which brings me to my last point …
        The “Croft” in Croft Spiral Sieve was originally intended as a placeholder (initially I had a hard time coming up with a succinct name, and when I first launched my website I said exactly that and also admitted that it was incredible hubris to put my name on the mathematical “object” I was studying. I had every intention of taking my name out of it, but frankly the thing went out of control. After spending 100′s of hours teaching myself html, SEO and launching my primesdemystified.com site (along with another site that has the most in-depth listing of number 30′s mathematical properties anywhere on the web) taking my name off (with scores of .jpg’s to edit, etc.) suddenly became a huge project (though one I’ve had every intention of carrying out, regardless). I knew full well that putting my name on the spiral would be contentious. James’s message has prompted me to expedite that project and I thank him for being a catalyst, albeit a) I have no intention of abandoning my explorations, and b) I’m still hopeful that someone like James, instead of condemning people like you and me, will take the time to give us productive feedback about the patterns themselves … Is that asking too much?

  6. James says:

    I just remembered my previous comment and decided to check up on it. Let me first state that my post was not in any way directed to you (admin). I think it’s great that people without a formal education in a subject find it interesting enough to do research, and I apologize for the tone of that comment. It was late, and it annoyed me when Tao’s blog was being riddled with “cranks.” You see, I had just read a decent amount of Miles Mathis material and was reading Tao’s blog to get the “ugh wtf” taste out of my mouth…and that is when I saw many il-informed comments that aggravated me and directed me here. I do not think you should stop doing math, and really apologize for being that immature to where I would even suggest that you do such a thing. While I do think that it is unlikely that any of you will find anything, I think it is unlikely that the vast majority of number theorists will either, and so “hell, it can’t hurt to try!”

    On a more general note:
    It seems that many people in this community feel that math is “impenetrable” to amateurs. I have to *somewhat* agree with this notion, but I disagree with this being unnecessary. You have to realize that there are specific protocols in doing math, and that these are just expected. Having explored a decent amount of amateur proofs, I can say the following are pretty much standard flaws:

    1) Inability or refusal to use Latex (or a derivative of Latex). This just makes stuff aggravating to read. While not a mathematician (yet), I can tell you that my professors are extremely busy between research and actual class-related activities. To expect them to read anything as convoluted as “int(sin(x^x(ln(x))+3/cos(y))))dx*wedge*dy” and expect them to keep reading is borderline disrespectful. While I’m on this topic, I may as well mention how their time constraints severely limit the amount of time they have to devote to reading (and correspondence with regards to) the proofs of untrained “mathematicians.” Students will always be their first priority, and I don’t think it is a stretch to say that this a bad thing.

    2) Introduction of a ridiculous amount of neologisms, rarely with a well-defined definition. Annoying and unprofessional.

    3) Grandiose claims/extended history of the problem supposedly put to rest are included in the proof. This is annoying and unwarranted. No one cares about your ego, do not massage it in front of us and expect to not be laughed at.

    4) Over-development of the trivial parts of the proof. If you are sending a paper to a PhD, you can bet that you are only wasting their time with this stuff. Get to the point.

    5) Under-development, or an illogical development of the non-trivial parts of the proof. At this point, a lack of mathematical training usually becomes blatantly obvious. If your results are based on a very simple method, yet the problem has had a pretty long history in the “unsolved” column, then you can bet that there is something you aren’t considering. Basically, if it were that simple, it would have been solved by now.

    6) Refusal to accept criticism as valid and arguing for the validity of your result. Even a PhD would be laughed out of the building (or a license!) if he did this.

    Now, I’ll admit, while looking for an area to specialize in I looked at many unsolved problems. If I thought I had found something interesting I would go to a professors office hours and ask something along the lines of: “I’ve been reading about X and think I may have stumbled on a different way to view the problem…but I’m sure there is something I’m not considering.” I would then thank him when, surely enough, there was something (however small) that I was not considering. I wouldn’t storm in and say “I’ve solved X! READ THIS!” and then argue why he was wrong when he informed me of any missteps in my reasoning.

    A little more on the impenetrable fortress that is math…I honestly hate to say this, but math is so deep (and our knowledge base is growing ever deeper) that a graduate degree, or at least a very profound gift and deep self-study covering a wide a range of topics, is almost necessary to contribute. A little assignment: go to arxiv and view a random (valid) paper. Check the references. These are people with PhD’s that have devoted their life to math, who have been surrounded by the latest advances in their fields for years, and who are generally exceptionally bright. And yet they generally have to consult with many other specialists! Often a relatively large amount of specialists. Thinking that one person without access to these minds/education/*environment* can produce something that these men, or possibly thousands of these men, could not is a little silly to me.

    Last, but not least, Ramanujan would not be dismissed as a crank. He wasn’t then, and he wouldn’t be now. This is just unfounded rhetoric. And comparing your abilities/situation to the abilities/situation of Ramanujan is borderline hilarious.

    I’m a bit busy at the moment, so I have to end here. When I get around to it I will point out some flaws in certain patterns posted, and why they don’t extend (as they must to be relevant). I just felt bad about the tone of the post I had submitted and had to unsay a bit of it.

    And a little food for thought:
    Just as you wouldn’t go to a witch-doctor to have an illness treated, just as you wouldn’t call your plumber to defend you in court, and just as you wouldn’t go to see a dentist about a heart problem, you wouldn’t ask an untrained mathematician for insight into the Riemann Hypothesis. Similarly, but more relevant to the gripe with the mathematical community: just as a doctor wouldn’t call a witch-doctor for medical advice, just as a lawyer wouldn’t call his plumber for legal advice, and just as a dentist wouldn’t ask a heart doctor for advice on performing root canals, you just have to accept that a mathematician isn’t inclined to talk math with people who either aren’t qualified or who aren’t the future of the field (i.e. their students). There will always be the rogue plumber who knows the law well enough to offer decent advice, but you have to agree that this fact of life is necessary in that it saves a lot more time that it wastes. If your ideas are good enough, the internet is an excellent medium to getting them noticed. As Howard Aiken once said: “Don’t worry about people stealing your ideas. If your ideas are any good, you’ll have to ram them down people’s throats.” So post these ideas. Eventually, someone will notice. If not, your proof is just flat out wrong.

    • James says:

      Edit: In comment “1)” should be “…don’t think that this it is a stretch to say this is a **good** thing.”

      Pretty big typo :)

      • James says:

        Edit of edit: “1)” should be “…don’t think that it is a stretch to say that this is a **good** thing.”

        Wow, I need to go to bed! My apologies.

    • James
      Thanks for your further comment and yes I understand the need for mathematical notation – makes for efficient communication amongst mathematicians. Yes I agree that mathematics is so well developed that the amateur is unlikely to make significant insights.

      As for flaws I am aware of some already and my time is probably as valuable as your own so I have not documented them. Happy to hear of comments/criticism when you have time.

      However whilst I am clearly am amateur I did study Pure Mathematics at an English University (1975). I have to admit it defeated me but that was for reasons other than the mathematics. My frustration is that I have found it difficult to find a text that allows me to bridge my knowledge with that required to read even some of the older papers and books. The combination of lack of mathematical formatting on Word Press and knowledge makes it easier to dabble in pictures than serious maths.

  7. James says:

    I was wandering why you seemed so different than the majority of relatively untrained people who take an interest in number theory. It appears to be your exposure to math (along with an obvious maturity). The tendency of a certain group of people to claim proofs/*tremendous* insight of/into classical unsolved problems, the display of ego in smothering ones name all over every idea one has, and the tendency to copyright ones ideas before they’re even developed (copyrighting ideas puts you in the inner ring of James’ Inferno) is something I do not see from you, which is the only reason I had to come back and correct my harshness. I feel that there is a lot of hope for you to develop, as your ego doesn’t appear to be holding you back from admitting that you are fallible. While I am apparently a good bit younger than you, since I am somewhat “in the system,” I still feel the need to give a word of encouragement.

    *****
    The two sides of the “system argument” are incredibly interesting to me, and today has been a horribly unproductive day, so I have to go on a (hopefully) brief rant (***** denoting the beginning and end in case you wish to skip). Mainly with regards to the arrogance that destroys an amiable relationship between professional and amateur mathematicians. But who knows where it will end up :). First, let me state that learning math is an extremely humbling experience. Many of mans greatest intellects have developed extraordinary insights, and when I realize some of the clever thought processes implemented I generally grin from ear to ear. Many of these men have spent entire careers developing one subject – and when some uneducated man surfaces and claims to unravel their work in a weeks time, I honestly feel somewhat disgusted. Don’t get me wrong, I would absolutely love for one of them to be correct. I’m all for accepting a minor setback in favor of a greater possible progress. But there is usually a lack of thought that goes into these proofs, and there is always some lack of consideration or understanding. As a rule, when I write a proof, I’m constantly thinking “how could this be wrong?” Unfortunately, it seems as though people with inflated egos on the line generally never question their reasoning. I can only imagine them reciting “how could this not be right?! I’m a genius!” to the tune of some beautiful symphony they composed once they walked out of their mothers’ birth canal.

    Let me preface my next statement by saying that I am by no means implying that every time one finds something they see as interesting that they should send it off to a PhD. But at least these mailings wouldn’t be immediately discarded if the experts didn’t get so many of them; those who are untrained yet have good discretion on what to send could possibly contribute to more meaningful ideas. It turns out that many people with a poor sense of self-worth, or a narcissistic disorder, etc. have taken movies such as Good Will Hunting as realistic depictions of “genius.” The truth of the matter is, genius is a romantic era myth. An interesting post I came across today on a message board (which happens to have a high percentage of what many would call “geniuses”):

    “Do you know why there are so few distinguished physicists of the late 19th and early 20th centuries? It is not because their work is particularly difficult, but because there were very few physicists at the time. Even today there are not even half a million in the world, but it is enough that no individual genius stands out. They have not ceased to be born, they have ceased to be significant.”

    For a while I bought into the whole “genius” idea too…much to my regret. I switched to a math major during my sophomore year in college. As many do, I soon got hooked on logic puzzles, and did quite well on them. This eventually led to me questioning how I compared with others, so I took a high range numerical IQ test that was well respected within the intelligence community. I got my scores back, and I’ll just say that I cracked 160. The worst mistake I have ever made in my life was to take these scores even remotely seriously- I thought I could get by in math with a total disrespect for the field. I occasionally thought my professors were wrong (when they weren’t), and I remained on my pedestal, oblivious to the fact that I was not only being a douche, but a completely wrong douche at that. I’ll save the transition and I’ll just say that I eventually realized that despite the level of genius I thought I had, I had to admit that the greatest minds in history were, not only by my standards, but by basic probability, much greater than my own. I began to assume I was wrong, and sure enough, I began to see the light. I began to respect my professors as experts, and I began to understand things on a much deeper level. But now I was faced with undoing a terrible treatment of my calculus sequence and linear algebra (where my motto was to study before the tests, and only go to class to take the tests) while taking other courses. I still had my 4.0, but my understanding was far from rigorous. I now accept that these fields are each so intricate that time is the only way to achieve mastery. Thinking any level of human genius is appropriate for dismantling a well developed field of human thought, developed through intense scrutiny from a large body of mathematical minded people, is absurd. I dismiss my former attitude as me being an immature kid, but I still cringe at the thought. This disgust with my former attitude is very likely why I posted somewhat nastily in the first place.

    And to anyone who takes their IQ as anything more than “some psychiatrist playing with numbers” needs to wake up. First of all, the concept was intended to address cognitive disabilities, not cognitive excellence. My favorite example: Richard Feynman was reportedly tested at a 121 IQ. Richard Feynman just so happened to be one of the most brilliant men who has ever lived. Meanwhile, what have Vos Savant and Langan ever accomplished other than an advice column, and a thesis on pseudointellectual jargon, respectively? IQ is a terrible attempt of people to quantify the unquantifiable.

    *****
    Now, addressing your comment:

    First of all, I love pictures/diagrams as well. At least to me, in most cases these are better for seeing the moving parts of a particular problem. My main gripe is that operations on them are usually extremely “icky,” so once an intuition is developed I transition over to notation. Also, I don’t feel they can be inspected quite as deeply, at least with present day mathematics (or perhaps my own limitations). For me, they give more of a relatively immediate, relatively shallow, understanding that helps in transitioning into a slower developing, deeper understanding.

    With regards to Latex, I highly recommend that you learn it. It is really not very difficult. It may be because I have a decent background in computer science, but once I use a symbol, it usually stays in memory. It is also extremely well documented. Once I started, I literally got addicted to it. It just looked so cool to finally be able to write some friggen math with my computer!

    As far as developing your abilities, there are so many resources on the web. I’d just say that, in particular, you need to take the right approach in developing extending your knowledge, and you need to take the time to work out problems to reinforce what you’ve learned. I generally look at the easier problems and work them out in my head (or write every few steps if I can’t). When I get to the harder problems where I have to actually understand any concepts involved, I take my time. If it’s a theoretical course, I’ll usually start every problem and just let them fester. Eventually a solution usually pops out. I’m not sure what approach you’ve taken in reacquainting yourself with mathematics, but I’m in a similar position in that I’m solidifying my foundation at the moment. I know it’s been a while since you had any formal math schooling, so I’ll just list my own approach in case it can be of any help. I will warn you that I have ADD, so it generally helps me to juggle a few things at once. This approach helps to make connections between different fields, but I can see where it could be a bit off-putting to someone who hadn’t seen a chunk of the material in a while.

    i) Probability, combinatorics, modular arithmetic. Basically, the goal here is to develop a feel for numbers in a general sense. Not really the functional sense, where there are an infinite amount of values in consideration. These were probably my biggest strengths going in, so my review was only a couple of days. Mostly boring as, though it wasn’t an incredibly deep review, most of it seemed instinctual.

    ii) Review some trig/geometry. The point here wasn’t to memorize, but to become “comfortable enough” in deriving what I needed. This is something I find easiest to develop with time. When I have a tough time with something, I usually remember it from then on. The goal was to get comfortable to the point where I didn’t have to go through this very often.

    iii) One variable calculus. I cannot overemphasize the need to understand optimization and related rates. Sure, you may not use these ideas as much, but they reinforce the involved concepts on relatively deep level.

    iv) Linear algebra. Obviously basic operations, getting into the more theoretical topics of vector spaces, **linear transformations**, bases, rank, **nullity**, etc. Eigenvectors/eigenvalues would be nice, but could be skipped for a bit. May as well visit them now though.

    v) Multi-variable calculus and differential equations. For (iii) and (v), I highly recommend “Schaum’s Book of 3000 Solved Calculus Problems” (I think that’s right). There are a few errors, but it’s a great way to test your absorption of the material.

    Now is where things get fun :). Understanding the theory behind what you’ve been using!

    vi) Abstract algebra/set theory/number theory. Groups, rings, fields, homomorphisms, isomorphisms, and so on. My favorite field.

    vii) Analysis (one variable). I would go slowly on this one. There are many tricks to pick up, and a lot of non-intuitive things going on that need to be examined. I would suggest downloading Tao’s lectures from his website for this one. There is a lot of commentary, and I cannot imagine a text better suited for a “noob.” Tao is brilliant in his ability to relay things to relatively “common” folk (actually, he’s just plain brilliant).

    viii) Complex variables/analysis. Beautiful subject. Absolutely…beautiful. Changed my philosophy towards math. Deepened my respect for the mental universe that subsumes our physical one. To avoid a rant, I’ll have to not discuss complex analysis :) By the way, Tao has another excellent series of lectures (pdf) on complex analysis on his UCLA website.

    ix) Multivariable analysis. Will entail a bit of general topology, but most (hopefully all) resources will develop this idea within the treatment of the main subject (the analysis). Basically, topology shouldn’t need to be developed separately for this course. Also, make sure to visit differential forms, which is another beauty. This should naturally be visited after Stokes theorem, but I have seen professors who do not do this. If the notes you use don’t explain differential forms, well, then a self-study is a must.

    x) Topology. Fun subject. There are many areas of topology, of course, but I’ve only formally visited general topology. I can’t really comment too much here.

    I’m sure I’m missing what many would consider to be important subjects. For example, I know what is generally referred to by “Stats 2” is not included. Maybe it’s because I find the subject boring, but I just cannot go back and do this. I like math, not consulting tables. Basically, these were/are just my focal points.

    Some excellent resources, many of which are free: https://sites.google.com/site/scienceandmathguide/dashboard2/mathematics
    My favorite of the links contained within the above is: http://hbpms.blogspot.com/
    I downloaded textbooks for just about every course imaginable. At least most (maybe all) of them are excellent reads. Wide array of topics ranging from basic algebra to differential geometry to knot theory.

    • Hi James. I understand you have some trouble with us amatours, and maybe you right.
      Please go to http://www.scribd.com/srento and read my “Primepatterns”. I here explain why there no speciel interesting in Ulam’s spiral, and I nearly don’t use math (Im an idiot to math). Is I then an exception that math has to be difficult or ?

      Søren

  8. Huenyk says:

    As an amateur number theorist, I did play with geometrical patterns but I have abandoned these since such patterns do not seem to help in generalisation. Now I only use Maxima symbolic software to generate such prime patterns. Here is an example:
    sum(sum( f(n-3*k,n-2*k,n-k,n,n+k,n+2*k,n+3*k)*(is(equal(primep(n-3*k)*primep(n-2*k)*primep(n-k)*primep(n)*primep(n+k)*primep(n+2*k)*primep(n+3*k),true^7),true)-unknown)/(true-unknown),k,1,n),n,450,1000);

    f(199,409,619,829,1039,1249,1459)+f(179,389,599,809,1019,1229,1439)+f(47,257,467,677,887,1097,1307)+f(7,157,307,457,607,757,907)+
    ignore negative primes below:
    f(-11,199,409,619,829,1039,1249)+f(-31,179,389,599,809,1019,1229)+f(-163,47,257,467,677,887,1097)+f(-379,41,461,881,1301,1721,2141)+
    f(-431,-11,409,829,1249,1669,2089)+f(-647,-227,193,613,1033,1453,1873)+f(-773,-353,67,487,907,1327,1747)+f(-1061,-431,199,829,1459,2089,2719)+
    f(-1069,-439,191,821,1451,2081,2711)+f(-1237,-607,23,653,1283,1913,2543)+f(-1583,-743,97,937,1777,2617,3457)

    • Thanks for that! Looks like something I should be aware of, although it will take me a while to digest.

      • Mats Granvik says:

        It should be easy to understand. Let me explain.
        Do you know how to make the Fibonacci numbers? 1+1 = 2, 1+2 = 3, 2+3 = 5, 3+5 = 8 and so on. Then do the Tribonacci numbers 1+1+1 = 3, 1+1+3 = 5, 1+3+5 = 9 and so on. Continue like that to infinity, Tetranacci, Pentanacci, n-anacci.
        Now instead of adding to positive number, wrap the result with minus. -(1+1) = -2, -(1+-2)= 1, -(-2+1) = 1, -(1+1) = -2, and so on. Do the same for Triboacci -(1+1+1) = -3, -(1+1+-3) = 1, and so on.

        Now make it more complicated, put all these NEGATIVE n-anacci numbers next to each other with one column for each n-anacci number. That is, the negative Fibonacci numbers in the third column, the negative tribonacci numbers in the fourth column and so on.
        If you divide the result with the row index and sum it up, you will get the natural logarithm of n. That is, for negative Fibonacci numbers you will get the logarithm of 3, for negative Tribonacci numbers you will get logarithm of 4, and so on.

        Now make it really complicated. Let the recursions for negative n-anacci numbers that I described, run in cross directions to each other as in here:
        http://en.wikipedia.org/wiki/Von_Mangoldt_function#Expansion_of_terms
        So that you get matrix T. Then repeat and divide by either the row index or the column index (both row index and column index goes because of the symmetry), sum it up and you have the von Mangoldt function which is the fundamental function in the fundamental theorem of arithmetic, next after the Möbius function and the Dirichlet inverse of the Euler totient function.

  9. Mats Granvik says:

    The matrix I described above is closely related to Riemann zeta function products that give this graph.
    http://i.stack.imgur.com/5LirM.jpg

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