The Spiral Of The Primes

by Ted Andromidas

When I presented the draft of our last discussion on prime numbers and the notion of indicative quota, to one of my closer collaborators, I was filled with a sense of satisfaction, and wonder, at having gotten a glimpse at what I thought was the idea of number, the generation and distribution of the prime numbers and their connection to the notion of indicative quota. She read it, looked up and said, not the “Ah-ha” for which I had so patiently waited, but “Yeah, so what?”

I was stunned! I sputtered: “What do you mean ‘SO WHAT?’? Are you confused.”

“No, not really.” she said. “I was confused for a moment when you tried to convince me that Eratosthenes and Earthshines was a clever play on words, until I realized that you screwed up the spell checker; AND, despite the fact that you then tried to convince me that there was some pedagogical significance to putting the footnotes out of order, rather than just sloppy editing, I do see how Eratosthenes’ method works; it’s just: What’s the significance of this to indicative quota? And, as a matter of fact, what’s the significance of this problem of prime numbers at all?”

I walked away baffled and disoriented. I thought it was all so clear.

If someone were to say that an “indicative” quota, i.e. a systematic approach to raising money, is just like any other idea of quota, that it’s just a number reached by adding the money raised in any given week, and that any change in the that quota is merely an process of adding or subtracting from that number, could we not characterize that as an “axiom of the system of quota”? Then couldn’t…

“You say to somebody, ‘Here is the axiomatic problem.’ Everybody in mathematics who has a terminal degree–which is what happens to you before they put you in a body-bag–knows the hereditary principle. Even Bertrand Russell knows the hereditary principle–or knew it, wherever he is today. Everybody knows that if you construct a logical system– and mathematics as usually defined is nothing but a logical latticework– everyone knows that if you start out with a system based only on axioms and postulates, and you develop only deductive theorems based on these axioms and postulates, that the entire latticework, which can never be closed, consists of nothing but echoes of the axiomatic assumptions with which you started. Therefore, if one of the axioms is false, the entirety of that field of knowledge collapses.

“An example: If you say that the only thing that exists in arithmetic is the integers, as counting numbers–that everything else is synthetic–therefore, so the argument goes, all mathematics must be derived from the counting numbers as the axiomatic foundation. So you start with an axiomatic counting system, 1 + 1, you construct that, and from that elaborated basis you must develop all mathematics. This is essentially what Russell and Whitehead demanded: radical nominalism. Therefore, as the case of prime numbers implicitly proves–the Euler-Riemann theorem, the work of Gauss on prime number sequences, the ingenious foresight of Fermat on this question, the work of Pascal on the question of differential number series–the entire history of mathematics, centering around this fantastic little problem of prime numbers…”  —Lyndon LaRouche, Schiller Institute Conference, 1984

If we look at the process of counting as iteration, as a function of a one dimensional manifold, we are confronted with “…this fantastic little problem of prime numbers…”: we can not determine the distribution or “density” of prime numbers between 1 and any given number N by any other means than that of Eratosthenes? We can not determine what the next prime number, or for that matter, any future prime number in a counting series, will be before it is actually generated by counting?

Begin counting 1, (2), (3), 4, (5), 6, (7)…;(all prime numbers are bracketed in parentheses) at first it seems that all the prime numbers are also all the odd numbers; that is by just adding 2 + one, then add 2 + 3, i.e. f(p)= (1+2x). We see, to generate the primes, but as we continue to count, (2), (3), 4, (5), 6, (7), 9, 10, (11), (13), 14, 15, 16, (17), (19), 20, 21, 22, (23) 2…,the “pattern” or function seems to change. For a while it seems to be f(p)= (6x +/- 1) till we reach (23), then it changes again. We seem unable to discover a successor function for, not just all the prime numbers, but any particular continuous series of prime numbers.

There have been innumerable functions, theorems, hypotheses, corollaries and conjectures written on this problem: The Prime Number theorem, the Reimann Hypotheses, the Twin Prime conjecture, the Goldbach Conjecture, and the Opperman Conjecture just to name a few.

Let us look at a conjecture referenced several times by LaRouche, that of Pierre de Fermat. Fermat conjectured that every number of the form (22^n +1) is prime. So we call these the Fermat numbers, and when a number of this form is prime, we call it a “Fermat prime”; the only known Fermat primes are the first five Fermat numbers: F0=3, F1=5, F2=17, F3=257, and F4=65537.

In 1732 Euler discovered 641 divides F5 and F(n) has been extended to 31, (i.e. 22^31 + 1) and no other primes have been generated by this function. It is, therefore, likely, yet not proven, that there are only a finite number of Fermat primes.

You might remember that the Fermat primes were the subject of a recent “Reimann for Anti-Dummies”. Gauss proved that a regular polygon of n sides can be inscribed in a circle with Euclidean methods (e.g., by compass and straightedge) if and only if n is a power of two times a product of distinct Fermat primes. (Hopefully we will look at this problem from the vantage point of Riemann’s and Dirichlet’s correction of Euler, if I can figure it out by then.)

Anyway, for now, let’s just continue to investigate the phenomenon of counting and the primes.

The Spiral of Prime Numbers

We saw last week that, using the method first developed by Eratosthenes in the 3rd Century B.C., with the circle as 2 dimensional manifold, we could construct a cyclical or modulo approach to the determination of the distribution of the prime numbers in the one dimensional manifold of the number line. The limitations of that approach are obvious. Moreover, it actually tells us more about the process of generating the non-prime, composite numbers of the number field as, implicitly, an ongoing succession of prime number cycles, than revealing something about the generation of the prime numbers themselves.

As a prime number is generated it is implicitly the modulus of an ongoing cycle, which intersects all past and future cycles of previous prime numbers, transforming the entire number field past and future. Yet, we seem unable to account for the generation of that singular event in the number field, the generation of a prime number, till, in fact, it occurs.

Perhaps it is the way we count; let us “count” differently. Rather than imagining the number line as straight, one dimensional manifold, i.e. 1, 2, 3, 4, 5, 6, 7…etc., is it possible to count in two dimensions? As we’ve seen from the dialogues of Philosph and Cando, numbers in a 2 dimensional manifold are not necessarily what they are in one; but rather than looking at the characteristic differences between two dimensional and one dimensional measure, let us take a more simple construction, and see what happens. Let us generate the number field in 2 dimensions by using a simple a kind of “Archimedean spiral”.

In the center of a piece of note paper write the number 0. To the right of that write the number 1; above one write 2. Now count to the left 3, 4. Below 4 write 5(at the same level as 0); below 5 write 6. To the right of 6 write 7, 8, 9; above 9, at the same level as 1 write 10, and go up 11, 12, 13; now count left of 13, 14, 15, etc. As you count this way, you will generate a spiral of numbers.

Now, beginning with zero, start counting the numbers spiralling out from there.(see figure 1). Try it; it is really not difficult.

(figure 1)

4-(3)-2 | | (5) 0–1 |


What do you notice, almost immediately: a certain number of prime numbers are generated along various diagonals of the number field.(see footnote 1)

Now, if we begin counting with 5 as our first number in the center of our spiral, we notice that between 5 and its square, 25, all the numbers that lie on a diagonal connecting 5 and 25 are prime numbers. They are not all the prime numbers between 5 and 25, but they do define a successor function of prime numbers between 5 and 25.

Begin counting at 11 and we generate a diagonal of prime numbers along the axis between the prime number 101 through 11, to 121, the square of 11. If we start counting with our Archimedean spiral at 41, we discover the same generating characteristic: it is a line prime numbers which stretch along the 41 and 1681 diagonal. In fact, if we count in this manner from 41 to 10,000,000, half of the numbers on that diagonal will be prime.

When we count in the one dimensional manifold, we can, through the sieve of Eratosthenes, determine that the cyclical, modulo characteristics of the counting numbers, the integers is ordered through the two dimensions of circular action. Yet, there seems to be no “connectedness” at all to the ordering characteristics of the prime numbers, no “pattern” seems to emerge.

When we actually begin to count in a two dimensional manifold, a “connectedness” emerges almost immediately, numerical shadows on the wall of Plato’s cave. Why? Is there some ordering principle from a higher, perhaps 3 dimensional manifold, ordering the two dimensions of our spiral counting? It doesn’t provide us with an actual function for determining the distribution of the prime numbers, nor does it help us develop a successor function, but it does impel us on to a notion of a succession of ordering principles, as we will begin to see in our next discussion.

And finally, to my collaborator’s insistent, “So what? What’s the significance of the prime numbers, anyway”, Karl Friedrich Gauss would reply:

“The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and prolix that even for numbers that do not exceed the limits of tables constructed by estimated men, i.e. for numbers that do not yield to artificial methods, they try the patience of even the practiced calculator… The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.”

— Karl Friedrich Gauss, Disquisitiones Arithmeticae (translation: A. A. Clarke)

Footnote 1.

Here is a list of the first prime numbers to aid you in your investigations.   2 3 5 7 11 13 17 19  23 29 31 37 41 43 47 53 59  61 67 71 73 79 83 89 97 101  103 107 109 113 127 131 137 139 149  151 157 163 167 173 179 181 191 193  197 199 211 223 227 229 233 239 241  251 257 263 269 271 277 281 283 293  307 311 313 317 331 337 347 349 353  359 367 373 379 383 389 397 401 409  419 421 431 433 439 443 449 457 461  463 467 479 487 491 499 503 509 521  523 541 547 557 563 569 571 577 587  593 599 601 607 613 617 619 631 641  643 647 653 659 661 673 677 683 691  701 709 719 727 733 739 743 751 757  761 769 773 787 797 809 811 821 823  827 829 839 853 857 859 863 877 881  883 887 907 911 919 929 937 941 947  953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693