Prime numbers and "prim" numbers

[Enola_Straight]

In this thread
http://boards.straightdope.com/sdmb/showthread.php?t=496985&highlight=prime
I pointed out how all primes are either 6x=1 or 6x-1 (note: that is NOT a generator…some 6x+/-1 numbers are not prime ;))

This guy has expanded the concept with what he calls “prim” numbers:

http://french.chass.utoronto.ca/as-sa/ASSA-14/article7en.html

123=6
1234=24
12345=120
123456=720
1234567=5040

and primes lie on the prim +1 and prim-1 lines

http://french.chass.utoronto.ca/as-sa/ASSA-14/modulus6-spiral90.jpg

Does his theory hold true?

[Wendell_Wagner]

First, note that the paper that you link to is not published in a mathematical journal at all. It’s published in a literary research journal called Applied Semiotics. Furthermore, note that the author of the article is also one of the two editors of the journal. Then look at the definition of “prim” that the article gives (or at least I think this is the definition):

> These may be loosely understood as those composite integers having the
> highest number of distinct factorizations in their immediately neighborhood.

I think that that is too vague to be a useful definition. I suspect that there isn’t enough definite mathematical content in this article for us to be sure about what it’s talking about. But there’s something even weirder going on. The link you gave us supposedly is to the seventh article in the fourteenth issue of Applied Semiotics. But here is a link to the table of contents of the fourteenth issue of Applied Semiotics. Note that there are only six articles in the issue:

http://french.chass.utoronto.ca/as-sa/ASSA-No14/index.html

I think therefore that this is the co-editor of the journal screwing around with the website of the journal so that he can insert his own article about a subject that he knows little about.

[Thudlow_Boink]

Thanks, Wendell; it didn’t look like a standard mathematical article to me.

For one thing, I noticed this excerpt:

Thus it is not surprising that the most highly divisible integers, the multiples of 6 (which incidentally make up the Assyrian and Babylonian systems for measuring time and angles that we inherited in the West) are adjacent to each and every occurrence of prime numbers greater than three. Stated otherwise, all integers adjacent to prims like multiples of 6 have zero factors and are therefore prime, unless by mechanical ‘providence’ they themselves have a prime by which they are divisible.

Which, as far as I can tell
(1) gives the same fact that the OP does, which, as the thread the OP linked to points out, is actually fairly obvious and non-profound
(2) includes a parenthetical remark that doesn’t seem to have anything to do with the rest of the article, and
(3) concludes that numbers adjacent to “prims” (like 6k-1 and 6k+1) are prime, except when they aren’t.

[Exapno_Mapcase]

An image search for Ulam’s Spiral brings up a plethora of very cool images.

That’s a deep result. From my quick read, the article’s figure 4, his big reveal, is a trivial representation of the 6X+1, 6x-1 observation. It has to be true. You showed that yourself.

And his question, “Why is it that integers having a maximum number of factors are immediately adjacent to integers with a minimum of whole factors?” is answered by observing it’s another way of saying exactly the same thing.

I think. I can’t say I tried very hard to read sentences like "Clearly, if a prim number n[sub]prim[/sub] is divisible by an integer factor greater than one, f, then the next occurrence of an integer that is also evenly divisible by f occurs at n[sub]prim[/sub] +f. Similarly, the nearest integer inferior to n that is evenly divisible by f must prim occur at n[sub]prim[/sub] -f. " at this hour.

[Thudlow_Boink]

@Exapno_Mapcase

I think. I can’t say I tried very hard to read sentences like "Clearly, if a prim number n[sub]prim[/sub] is divisible by an integer factor greater than one, f, then the next occurrence of an integer that is also evenly divisible by f occurs at n[sub]prim[/sub] +f. Similarly, the nearest integer inferior to n that is evenly divisible by f must prim occur at n[sub]prim[/sub] -f. " at this hour.

That sentence makes sense to me, even though it’s not very profound. What it’s saying (IIUC) is that if a number (say, 432) is divisible by another number (say, 6), then you have to go up to 432+6 or down to 432-6 to find another nearby number that’s also divisible by 6.

[ultrafilter]

@Thudlow_Boink

That sentence makes sense to me, even though it’s not very profound.

It’s not just not profound; it’s so obvious that anyone who thinks it’s worth saying clearly doesn’t have any sense of what makes an interesting mathematical statement. That doesn’t exactly fill me with confidence regarding the article.

[Enola_Straight]

I figured, if these ever larger “prim” numbers do indeed weed out non-primes (note, I do not say find primes) then could they not speed up the search for ever larger primes?

If (1234)+/-1 weeds out non-primes …starting with 24… four times faster than
(123)+/-1, and (12345)+/-1 weeds out primes five times faster than (1234)+/-1, wouldn’t this mean the speed in prime searching would increase geometrically?

Write a program capable of “switching gears” when a prim number is reached?

UNLESS…6x+/-1 catches primes larger prims miss…:dubious:

[Enola_Straight]

I just checked…24+/-1 indeed misses primes between 24 and 48.

[Wendell_Wagner]

Enola Straight, where did you even find out about Marteinson’s article? Apparently he’s been trying to persuade mathematicians to look at his ideas and has been getting shot down in every forum he tries:

Do Infinitely Many Prime Pairs Exist?

http://bbs.slsee.com/viewthread.php?tid=61649

He apparently is a smart knowledgeable guy who’s convinced that he’s smarter and more knowledgeable than he actually is (a type that I’ve met before):

http://french.chass.utoronto.ca/as-sa/editors/pgm.htm

He’s been bouncing around in miscellaneous teaching jobs for a decade or two. He thinks it’s important to know that he can trace his ancestry back more than a thousand years in Iceland.

[Great_Antibob]

@Enola_Straight

I just checked…24+/-1 indeed misses primes between 24 and 48.

Of course it misses primes.

In these cases, it helps to know WHY things work.

All primes (except for 2 and 3) are of the form “6k +/- 1” because 2 and 3 are prime (which is why they are the exception).

So, 6k, 6k+2 and 6k-2 are all divisible by 2. 6k+3 is divisible by 3. That only leaves 6k+1 and 6k-1, which are the only numbers that can be (but are not always) prime. If they are not prime, they are divisible by a prime larger than 3. For example, 25 is 64+1 and divisible by 5 (larger than 3). And 35 is 66 - 1 and divisible by 5 and 7 (both primes larger than 3).

If we extend this notion, we actually go up to 30k (not 24k) +/- various values, because the next largest prime after 2 and 3 is 5.

From that list, we can eliminate 30k+any even number (divisible by 2). Likewise, we can eliminate 30k+{3,9,15,21,27} because they are divisible by 3. Further, we can eliminate 30k+{5,25} because they are divisible by 5.

That leaves 30k+/-1, 30k+/-7, 30k+/-11, 30k+/-13.

So, all primes larger than 5 can be expressed in one of the forms {30k+/-1, 30k+/-7, 30k+/- 11, 30k+/-13}.

Notice that any of these primes can also be expressed in the form 6k+/- 1.

For example:

 30k+1 = 6*(5k) + 1 = 6*m + 1. 
 30k+7 = 6*(5k) + 6 + 1 = 6*(5k+1) + 1 = 6*n + 1
 30k + 11 = 6*(5k) + 12 - 1 = 6*(5k +2) - 1 = 6*p - 1
 30k + 13 = 6*(5k) + 12 + 1 = 6*(5k + 2) + 1 = 6*p + 1

I know that 8 possibilities is not as exciting or concise as 6k+/-1, but that’s where the math takes you.

Naturally, we can extend this further (the next step would be 210k+/- m), but it becomes a much less useful guide at that point.
BTW: this is NOT a particularly good way of detecting primes. It’s a good way for detecting COMPOSITES. If you have a prime candidate, you still have to determine whether or not it is divisible by a smaller number. For example, if a number can be expressed as 6k+3, you automatically know it’s NOT prime. But if a number is of the form 6k+1, it might be prime or composite. You still have to do more checking to determine primality.

[Tom_Tildrum]

@Enola_Straight

123=6
1234=24
12345=120
123456=720
1234567=5040

Aren’t these factorials? I mean, doesn’t this group of numbers already have a name?

[Exapno_Mapcase]

@Tom_Tildrum

Aren’t these factorials? I mean, doesn’t this group of numbers already have a name?

Correct. And factorials necessarily are, after 2 and 3, of the form 6X.

[Thudlow_Boink]

@Tom_Tildrum

Aren’t these factorials? I mean, doesn’t this group of numbers already have a name?

One of the problems with the article is that I had to struggle to figure out what his definition of “prim number” is, and I’m still not sure. The article says that they’re multiples of factorials:

An exhaustive investigation of such possibilities between 1 and 500 suggests that only certain regular, rectangular numbers, which I call ‘prim’ numbers because they are ‘prim and proper,’ or highly divisible, give prime numbers simply by the addition or subtraction of one. These may be loosely understood as those composite integers having the highest number of distinct factorizations in their immediate neighborhood. Further investigation reveals that the prim or most highly divisible numbers are the multiples of the factorials, 1.2.3, 1.2.3.4, 1.2.3.4.5, and so on. Thus we may regard the multiples of six as highly divisible and prim, the multiples of 24 as even more highly divisible and more prim, and so on with the multiples of 120, 720, 5040, etc.

But I can’t tell whether he’s defining “prim numbers” to be numbers that are multiples of factorials, or to be numbers that are “highly divisible” (which is too vague to serve as a definition), or to be numbers that “give prime numbers simply by the addition or subtraction of 1” (which would not be consistent with them being multiples of factorials, since, for example, 24+1 is not prime).

[Great_Antibob]

I’ll admit I can hardly read a paragraph of the paper without being very disappointed.

Besides the poor writing (there’s no real definition or work shown), there’s not much that’s interesting or profound, as ultrafilter has already pointed out.

It’s not profound that so-called “prim” numbers have lots of factors. But the fact that the author doesn’t recognize WHY this should be the case is a major cause for concern.

Also, his potential approaches for solving the Goldbach Conjecture or Riemann Hypothesis are asinine. I’ve seen better attempts by non-academics and high schoolers.

The Goldbach one amount to “if we could prove this other thing that is equivalent to the conjecture, then we could prove this conjecture”.

The proposed Riemann approach is worse. It basically is not actually an approach and just strings some words together. And the words apparently (his language is not precise here) involves shifting from the distribution of primes to the distribution of ‘prims’, for which he also provides no substantive work.

[Saint_Cad]

My understanding is that “prim” is a number that has so many factors that prim+1 or prim-1 is prime. For example 7! is divisible by 2, 3, 4, 5, 6 and 7 so 7!+1 cannot be divided by any of those numbers. Sounds like a wonderful prime generator until it fails miserable which is taught in sixth grade.

My favorite part

A Simpler Algorithm
To find all primes up to n, we therefore need only:

a) stop at each multiple of 6, m
b) consider m+1 and m-1 as candidates
c) test each candidate by dividing it by each prime ≤√m
d) conclude a candidate is prime if all such tests show a non-zero remainder.

This is 7th grade math

There is nothing in this paper that a bright (not even gifted) middle-school student interested in number theory could come up with on their own in an hour. If you are interested in the author’s background, here you go. And yes he has a Ph.D.

[nivlac]

@Saint_Cad

… If you are interested in the author’s background, here you go. And yes he has a Ph.D.

But not in math! What a surprise.

[KarlGauss]

@nivlac

But not in math! What a surprise.

It’s in “Comedy”, actually. I kid you not!

[Wendell_Wagner]

Saint Cad writes:

> If you are interested in the author’s background, here you go.

Um, Saint Cad, didn’t you notice that I already gave a link to that same URL?

[Enola_Straight]

@Wendell_Wagner

Enola Straight, where did you even find out about Marteinson’s article? .

I just goggled “6x+1” “6x-1”.
Looking again at this image
http://french.chass.utoronto.ca/as-sa/ASSA-14/modulus6-spiral90.jpg
I noticed that some of the composite non-primes were divisible by 5, so I figured, since primes tend to thin out the deeper you look, you needed to weed out non-primes more efficiently.

(23)+/-1
(235)+/-?
(2357)+/-?
.
.
.
.
(235*…*n)+/-?
With ever larger “weeders” (what the OP in my link referred to as “prim” ) one could quickly look deep along the number line; upon finding one, backtrack with the next smallest weeder to catch “missed” primes.

[Enola_Straight]

Here’s an idea:

According to Gauss the “prime density” up to a certain number (how many primes up to a hundred, up to a thousand, etc) is approximate to the logarithmic interval.

What function approximates the “jagged staircase” after you weed with 6x+/-1?