Prime fractions

[robert44444uk]

Here is a curiosity I do not totally understand:

For primes 2,3,....p(n), calculate all fractions p(1)/p(2) where p(2)>p(1), order the fractions, and calculate the differences between ordered pairs.

We may ignore the differences between the 0 and the first ordered fraction and the last ordered fraction and 1, as 0 and 1 are not in the set of prime fractions. The first ordered value corresponds to 2/p(n), and the last ordered value is 1-[p(x)/{p(x)+2}] where p(x) is the smaller prime in a twin, but a member of the largest twin pair less than p(n)


What seems curious is that the largest differences seem to straddle well known landmarks in the line from 0 to 1.

With n=46 and p(n)=199, providing 1035 fractions = n/2*(n-1) and the largest difference is between 0.32960894 (decimal representation simplified to 8 sig places) and 0.33668342, straddling approx. 1/3 and the next largest are at:

0.19745223 and 0.20353982, straddling approx. 1/5
0.79775281 and 0.80346821, straddling 4/5
0.49720670 and 0.50259067, straddling 1/2
0.92746114 and 0.93258427, straddling 93/100

The system appears to break down after these large differences. For example, the next largest difference:

0.74193548 and 0.74647887, straddles 0.7442072
but then 0.08900524 and 0.09352518, straddles 1/11
1/7 is the straddle for the 11th largest difference
1/13 is the straddle for approx 20th largest difference

Maybe I am reading too much into this, but maybe someone can help explain.

I would be useful for a programmer to run many series of fractions, with increasing n to see where the largest gap lies and if it straddles a well known landmark.

Running the first few n

n | #fractions | lower | higher | straddle median | close to
3 3 0.4 0.6 0.5 1/2
4 6 0.428571 0.6 0.5142855 ?? 1/2
5 10 0.4545454 0.6 0.5272727 ?? 1/2
6 15 0.714285714 0.846153846 0.78021978 ?? 4/5
7 21 0.294117647 0.384615385 0.339366516 1/3
8 28 0.454545455 0.538461538 0.496503497 1/2
9 36 0.304347826 0.368421053 0.336384439 1/3
10 45 0.304347826 0.368421053 0.336384439 1/3
11 55 0.47826087 0.538461538 0.5083612043 1/2

Further information on prime fractions and my thoughts can be found at:

http://tech.groups.yahoo.com/group/p.../message/18754

Regards

Robert Smith

[R.D. Silverman]

Originally Posted by robert44444uk

Here is a curiosity I do not totally understand:

For primes 2,3,....p(n), calculate all fractions p(1)/p(2) where p(2)>p(1), order the fractions, and calculate the differences between ordered pairs.

"All fractions p(1)/p(2)" consists of a single fraction: 2/3

We may ignore the differences between the 0 (sic) and the first ordered fraction

You have not defined the expression "ordered fraction".
Do you mean the first fraction in the ordered set?

and the last ordered fraction and 1, as 0 and 1 are not in the set of prime fractions. The first ordered value corresponds to 2/p(n),

This is gibberish. What does "corresponds to" mean? You have not
defined n, so how can 2/3 "correspond" to it?
and the last ordered value is 1-[p(x)/{p(x)+2}] where p(x) is the smaller prime in a twin, but a member of the largest twin pair less than p(n)

What seems curious is that the largest differences seem to straddle well known landmarks in the line from 0 to 1.

More gibberish. The word 'straddle' is meaningless; you have not defined it.
"well known landmarks" is mathematical nonsense. WTF is a "landmark"???

With n=46 and p(n)=199, providing 1035 fractions = n/2*(n-1) and the largest difference is between 0.32960894 (decimal representation simplified to 8 sig places) and 0.33668342, straddling approx. 1/3 and the next largest are at:

Now the discussion has totally devolved into numerology.

There may be something here worth studying. But since you
have not made any mathematically meaningful statements, it is hard to
know. You post consists of poorly defined statements and notation and
is essentially just codswallop.

This post would rank pretty high on my Crankometer.

[cheesehead]

Originally Posted by R.D. Silverman

"All fractions p(1)/p(2)" consists of a single fraction: 2/3

Did you consider the possibility that the OP meant

"all fractions p_n1/p_n2 where p_n2>p_n1" ?

Of course, that (interpreting the difficulty with the post as being merely a simple notational one, rather than its content being numerological gibberish) could render most of your criticism moot and lower your Crankometer reading, but failure to consider reasonably simple alternative interpretations has rarely stopped you before.

By the way, if you'd bothered to follow the link http://tech.groups.yahoo.com/group/p.../message/18754, you'd have seen a notation less likely to be misinterpreted.

[robert44444uk]

Thank you, Bob, as always, for replying.

Sigh! I had forgotten that you lurk to make precise those statements I have made after a glass of wine or two. (And that includes split infinitives, I forecast)

But, to make things a bit clearer (I hope):

p(n) is the nth prime

p(1) and p(2) - did not mean this, I meant as Cheesehead describes, that is, to describe a set of fractions created from all of the primes up to p(n), a smaller prime as the numerator of any member and a larger one as the denominator.

For example, for n=4, then p(n)=7

The set of 6 fractions is 2/3, 2/5, 3/5, 2/7, 3/7, 5/7

And in order of size they are 2/7, 2/5, 3/7, 3/5, 2/3, 5/7

Or stated approximately in fractions:

0.285714286,0.4,0.428571429,0.6,0.666666667,0.714285714

And their differences are approx:

0.114285714, 0.028571429, 0.171428571, 0.066666667, 0.047619048

The largest difference is 0.171428571, or more precisely (3*7-3*5)/(5*7) or 6/35

A straddle I am defining in terms of 3 values on a line from 0 to 1, the first is the smaller of two fractions in the ordered set of fractions, the third is the larger of the two, and the second, any point in between the first and the third. I am saying that points 1 and 3 straddle point 2.

In the simplified example, the largest difference, comprising two points on the line at 3/7 and 3/5 straddle some value between the two, but more specifically I am saying that it straddles 0.5, because that is the approx mid point. In other words, 3/7 and 3/5 straddles 3/6. I appreciate you might not like the terminology and I would appreciate alternatives.

Corresponds to: equal to, now that n is defined.

Bob: You have not defined WTF :)

I agree that "well known landmarks" is colloquial. To be more precise we might us the phrase "fractions with small numerators and denominators"

Regarding crankometer - I think that prime fractions are under-researched, maybe because there is no real use that I can see. But I like under-researched areas.

JK Andersen has demonstrated the following for small differences:

Let a<b and c<d be primes with a/b close to c/d.
Multiplying by bd gives that the two semiprimes ad and bc are close.
Let ad be the smallest (swap variables otherwise).
Then bc = ad + n for small positive n.
Dividing by bd gives a/b = c/d + n/(bd).
The smallest possible n is n=1. Then either bc or ad is even, so a or
c is 2, and then a/b and c/d are small. The difference 1/(bd) becomes
very small if the semiprimes are very large.

[R.D. Silverman]

Originally Posted by robert44444uk

Thank you, Bob, as always, for replying.

Sigh! I had forgotten that you lurk to make precise those statements I have made after a glass of wine or two. (And that includes split infinitives, I forecast)

But, to make things a bit clearer (I hope):

p(n) is the nth prime

p(1) and p(2) - did not mean this, I meant as Cheesehead describes, that is, to describe a set of fractions created from all of the primes up to p(n), a smaller prime as the numerator of any member and a larger one as the denominator.

For example, for n=4, then p(n)=7

The set of 6 fractions is 2/3, 2/5, 3/5, 2/7, 3/7, 5/7

And in order of size they are 2/7, 2/5, 3/7, 3/5, 2/3, 5/7

Or stated approximately in fractions:

0.285714286,0.4,0.428571429,0.6,0.666666667,0.714285714

And their differences are approx:

0.114285714, 0.028571429, 0.171428571, 0.066666667, 0.047619048

The largest difference is 0.171428571, or more precisely (3*7-3*5)/(5*7) or 6/35

A straddle I am defining in terms of 3 values on a line from 0 to 1, the first is the smaller of two fractions in the ordered set of fractions, the third is the larger of the two, and the second, any point in between the first and the third. I am saying that points 1 and 3 straddle point 2.

In the simplified example, the largest difference, comprising two points on the line at 3/7 and 3/5 straddle some value between the two, but more specifically I am saying that it straddles 0.5, because that is the approx mid point. In other words, 3/7 and 3/5 straddles 3/6. I appreciate you might not like the terminology and I would appreciate alternatives.

Corresponds to: equal to, now that n is defined.

Bob: You have not defined WTF :)

I agree that "well known landmarks" is colloquial. To be more precise we might us the phrase "fractions with small numerators and denominators"

Regarding crankometer - I think that prime fractions are under-researched, maybe because there is no real use that I can see. But I like under-researched areas.

JK Andersen has demonstrated the following for small differences:

Let a<b and c<d be primes with a/b close to c/d.
Multiplying by bd gives that the two semiprimes ad and bc are close.
Let ad be the smallest (swap variables otherwise).
Then bc = ad + n for small positive n.
Dividing by bd gives a/b = c/d + n/(bd).
The smallest possible n is n=1. Then either bc or ad is even, so a or
c is 2, and then a/b and c/d are small. The difference 1/(bd) becomes
very small if the semiprimes are very large.

You have not stated any problem!!! What question are you
trying to answer???

You have ordered those rationals in (0,1) consisting of only prime
numerator and denominator up to some bound. You have computed
successive differences between the terms. But I still see no question
to be answered.

[xilman]

Originally Posted by robert44444uk

Bob: You have not defined WTF :)

A clear omission on Bob's part. Please allow me to assist him.

"WTF" is the universal interrogative participle. It may be used to replace a number of other interrogatives such as "what", "who", "why", "which", "how" and so on.

Consider the parallel whereby a pronoun such as "it" may be used to replace any of a number of nouns such as "dog", "grammar", "music" and, for that matter, "'WTF'" --- note the appearance of quotation marks and so acts as a noun.

Hereby ends today's grammar lesson.


Paul

[bsquared]

Originally Posted by xilman

"WTF" is the universal interrogative participle...

A very polite way to put it :-)

[robert44444uk]

Originally Posted by R.D. Silverman

You have not stated any problem!!! What question are you
trying to answer???

It would be useful for a programmer to run many series of fractions, with increasing n to see where the largest gap lies and if it straddles a well known landmark.

Above from my original post.

It is evident that the largest difference can be defined as a fraction, because, for position 1 =a/b and position 3 =c/d, (a,b,c,d prime, a<b, c<d) then the difference is simply (cb-ad)/bd and the mid point twixt the straddle values is exactly a/b+(cb-ad)/2bd = (ad+cb)/2bd

So the first few largest differences for increasing n are:

n|members of set|posn 1|posn 3|mid point| mid point as a fraction|(also straddles)
3 3 0.4 0.6 0.5 1/2
4 6 0.428571 0.6 0.5142855 18/35 (1/2)
5 10 0.4545454 0.6 0.5272727 29/55 (1/2)
6 15 0.714285714 0.846153846 0.78021978 71/91 (4/5??)
7 21 0.294117647 0.384615385 0.339366516 75/221 (1/3)

In the case of n=4,5 the mid points are 18/35 and 29/55, but these also straddle 1/2. In the case of n=7, although the mid point is 75/221, the positions 1 and 3 (end points) also straddle 1/3.

So I am interested to find out if, for larger n, the end points of the largest difference straddle relatively simple fractions.

I want to find out if the differences which appear "out of line" (for example n=6) only occur for small n. (although n=6 straddles 4/5 this is hardly remarkable, it also straddles 3/4)

Only by testing to large n can I find this out. If I had any mathematical software then I would investigate further, but I do not.

I would be nice to hypothesise that there are limited number of simple fractions that straddle the end points of the largest differences for all n, but this is obviously not reasonable to hypothesise from such a small sample.

Remember that for n=46, p(n)=199, the median value of the largest gap is quite close to 1/3, the second largest gap median is quite close to 1/5 etc.

[R.D. Silverman]

Originally Posted by robert44444uk

Above from my original post.


<snip>

So I am interested to find out if, for larger n, the end points of the largest difference straddle relatively simple fractions.

Since you have failed to define "relatively simple" in a way that can
be quantified, the question is meaningless.

Go look up the word "height" in relation to the height of fractions.

It is clear that as n-->oo, the interval that has the largest distance
will NOT contain fractions of small height, but it depends on the word small.

Try injecting a little rigor into your discussion. One can give asymptotic
estimates for the length of the longest interval as n-->oo. One can then
give an estimate of the logarithmic height of rationals contained by that
interval. It is clear that for most large n (most has a well defined meaning
here in terms of measure) the longest interval will not contain a fraction
whose height is bounded by a fractional power of log(n). This is easily shown
simply by counting the number of fractions whose height is bounded by
log(n).

What are you *reallY* trying to ask? You still have not asked a question that
can be posed in any meaningful, rigorous way.

And if one interprets your question about "simple fractions" in the broadest
way possible, then the answer is clearly NO. Most of the time the largest
interval will not contain a "simple fraction". But this depends on YOUR
meaning of "simple fraction" in relation to n. Please give an exact
definition, as a FUNCTION OF n, what "simple" means.

[ewmayer]

Originally Posted by xilman

Hereby ends today's grammar lesson.

WTF??

[Housemouse]

We all know from the Wendy's commercial that WTB =where's the beef?

WTF=where's the fries?