20080826, 13:59  #1 
Jun 2003
Oxford, UK
3604_{8} Posts 
Prime fractions
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*(n1) 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 
20080826, 14:52  #2  
Nov 2003
2^{2}×5×373 Posts 
Quote:
Quote:
Do you mean the first fraction in the ordered set? Quote:
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) Quote:
"well known landmarks" is mathematical nonsense. WTF is a "landmark"??? Quote:
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. 

20080826, 14:57  #3  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
Quote:
"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. Last fiddled with by cheesehead on 20080826 at 15:12 

20080827, 12:18  #4 
Jun 2003
Oxford, UK
2^{2}·13·37 Posts 
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*73*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 underresearched, maybe because there is no real use that I can see. But I like underresearched 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. 
20080827, 12:36  #5  
Nov 2003
2^{2}×5×373 Posts 
Quote:
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. 

20080827, 16:09  #6 
Bamboozled!
"ð’‰ºð’ŒŒð’‡·ð’†·ð’€"
May 2003
Down not across
3·3,529 Posts 
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 Last fiddled with by xilman on 20080827 at 16:13 Reason: Correct an endash to an emdash. Doubtless there are other grammatical errors not yet corrected. 
20080827, 16:12  #7 
"Ben"
Feb 2007
2×5×337 Posts 

20080827, 17:25  #8  
Jun 2003
Oxford, UK
2^{2}·13·37 Posts 
Quote:
Quote:
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 (cbad)/bd and the mid point twixt the straddle values is exactly a/b+(cbad)/2bd = (ad+cb)/2bd So the first few largest differences for increasing n are: nmembers of setposn 1posn 3mid 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. Last fiddled with by robert44444uk on 20080827 at 17:41 Reason: Why not! 

20080827, 18:19  #9  
Nov 2003
2^{2}·5·373 Posts 
Quote:
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. 

20080828, 04:25  #10 
∂^{2}ω=0
Sep 2002
RepÃºblica de California
10110101010110_{2} Posts 

20080828, 12:36  #11 
Feb 2008
2^{5} Posts 
WTF ?
We all know from the Wendy's commercial that WTB =where's the beef?
WTF=where's the fries? Last fiddled with by Housemouse on 20080828 at 12:36 
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