Putting Dropbox on my computer has at last given me the chance to share the large variety of my amateur art works: fiction, non-fiction, drawings, photographs, music, scores, and even a silent movie. In other personal news of note, I offered NASA a haiku which was included in a DVD aboard the Maven mission to Mars which launched recently. It reads as follows:
Red world of dread war
and monstrous green mirages—
sighs of rusty stones.
A whole new section beginning to present material concerning the Sinus Medii (Central Bay) region of Earth's Moon has been added with a list of recognized features, and pointing out a class of previously unrecognized features. Images from the region photographed by the Lunar Orbiter missions have also been added with an original scan and a processed version to enhance detail. Research is continuing with regard to Earth's moon, so I expect to add a lot more material eventually.
The most recent addition is Volume 2 of the Apollo 10 images. Volume 1 was posted last month. My latest piece of music is for strings, The Böd, and that, plus its score, was the prior post. Over the next weeks I will finish assessing 50 million prime integers and will accordingly make a few changes adding some information to the short silent video The Story of Prime Numbers ... So Far.
In my free time I am also continuing to process prime integers to find their differences and compiling statistics on their distribution and abundance in search of a theater of replete numbers to accommodate values of twenty or thirty million. The largest theater of repleteness found so far, beginning at approximately 11,265,887 will only accommodate values up to about five and three-quarter million. So far 48 million primes have been assessed.
Let me take this opportunity to explain the concept of replete numbers which I have discovered through the cataloguing of prime integers, presented in greater detail in the silent video The Story of Prime Numbers ... So Far, linked under the Non-Fiction category below. Differences among prime numbers do not generally occur among the more familiar smaller numbers in numerical sequence. The first prime is 2, the only even prime, and the second prime is 3, forming a difference of 1, the only odd difference among primes.
The third prime number is 5, the only appearance of this odd number as a final digit among all primes, forming a difference of 2. This difference follows on 1, so all is well so far. The next prime is 7, also forming a difference of 2, and the next prime is 11, forming a new difference of 4, which follows 2, so all remains well. The next new difference to appear is 6 at the prime 29, so all continues to be well and the next new difference is 8 which is formed at the prime 97.
At the prime 127, however, something unexpected occurs when the next new difference formed is 14. The difference of 10 is not formed until the prime 149, and the difference of 12 does not occur until the prime 211. To describe that property of the region from 127 to 211, I have given the name irrepleteness. There are other irreplete zones of the number line as well beginning at the prime 541 where the difference of 18 is first encountered, (16 is not given until 1,847) and persisting beyond 1,361 where the difference of 34 is first found, until 5,623 where the difference of 32 is finally given.
The number line continues in good order through 9,587 where the next new difference of 36 is encountered, but then at 15,727 the next new difference of 44 is given. The irregularity becomes more acute at the prime 19,661 where 52 is given as a difference, more acute still at 31,469 where a difference of 72 is given, more acute yet at the prime 188,107 where a difference of 78 is found, and the differences persist in increasing irregularity through 4,652,507 where 154 is the difference. Finally at 13,626,407 repleteness is restored where the value of 150 is given as the difference.
The restoration is brief, however, for at 17,051,887 the next new difference found is 180. So far I have not been able to find a new replete region and the problem will require a great deal of diligent searching because a relatively large difference of 248 is unexpectedly encountered at the prime 191,913,031. My search so far has taken me through 48 million primes to 941,083,981 and one new difference was found at 895,858,267 which posted 228. Repleteness is not soon at hand, however, because 436,273,291 has already posted a difference of 282, raising the bar. If the bar is not raised again, that is if a nearby prime does not unexpectedly post some difference larger than 284, thirteen values remain to be posted to achieve repleteness.
The significance of repleteness lies in plotting graphs of equations, because all gradations of value cannot be registered where there are gaps among the differences of primes. Using values outside of replete zones induces gaps within the number line for all infinitesimals between each integer that can be expressed as a fraction with a denominator of any prime or its multiple which posts that missing difference. For instance, invoking values between 97 and 149 means that 149ths and 211ths, and all their multiples, are not available as values between the integers. Also not appearing are fractions of other primes with differences of 10 or 12, for instance: 191sts, 251sts, 223rds, and 293rds. Between 149 and 178, those with differences of 10 are restored, but those with differences of 12 are still lacking. This lack of registration produces spikes in the curvature of graphs, slopes of lines that cannot be resolved through any improvements of measurement. It is vital, therefore, to produce graphs that correspond to real events in nature be performed only with replete numbers for the measurements and estimates. (Tallies and constants are not affected.)
Such spikes of graphing found in physical investigations have been considered to represent actual properties of material things under investigation, but the discovery of the property of repleteness opens up the prospect that the loss of correspondence in results between calculation and experiment is actually a property of the numbers employed in making calculations. This is to say that the continuity of the number line can only be guaranteed within zones of repleteness. The number line is not an indifferent tool with uniformly infinite properties. The use of it as a quantifier is necessarily a sampling to some limit and its discontinuities are not consistent, but are a function of that sample.
Graphing those primes which post new differences together with those differences in both orders draws two lines of representation. There is close correspondence between 0 and 29 where the graph of the differences falls slightly below that of the primes. Then at 97 the graph of differences rises over that of the primes, indicating irrepleteness. It falls below again at approximately 178, indicating repleteness has been restored which is confirmed at 211 where the missing difference of 12 is reported. At present I do not know a method for calculating exactly where a zone of repleteness begins after the first one, so I have offered only an eyeball approximation, but zones of repleteness end exactly where a prime posts a difference larger than the sequence of even integers so far given.
Attempts to draw graphs of curvature for behaviors of physical phenomena will find the curvature intercepted by a given slope, depending upon the smallest value employed in measurement calculation. For instance if the value lies between 97 and 127, the slope will be 1 over 97, or 0.590678139310127°; if the values lies between 127 and 149 the slope will be 1 over 127, or 0.451147870181751°; if the value lies between 149 and 178, the slope will be 1 over 149, or 0.384535432973707°.
Research so far has established only these zones of repleteness: 0 to 97, thereby making small constants valid; 178 to 541, valid for calculations not exceeding 363; 4,609 to 15,727, valid for calculations not exceeding 11,116; and (approximately) 11,265,887 to 17,051,887, valid for calculations not exceeding 5,786,000. The technique to be employed for replete numbers in calculation is to add the value beginning the zone to all measurements and estimates, then to subtract it out in reporting the data labels. The results will give curves replacing spikes in data graphs, subject only to resolutions of measurement, which is a separate issue.
It is certain, however, from this observation that after some very large number has been invoked, the repleteness of the number line becomes a permanent feature for larger values and its full continuity can be forever guaranteed. This is because the total abundance of primes within the number line is given at nearly 29.5% by the Sieve of Eratosthenes, but the abundance of primes among the first million integers, where they are most abundant, is only a paltry 7.85% (approximately) and declines thereafter. The current total abundance of the region I have researched to 48 million primes stands less than 5.1%. In order for the total abundance to be achieved, it is necessary that there exist a final limit to the differences among primes, though not to the primes themselves. Once this limit, and every smaller even integer has been given as a difference, repleteness becomes a property of the entire infinite extent of the number line and its full continuity is guaranteed.
This realization has implications for cosmology because it shows that until the universe in its formation had realized that very large number, considering its values as an immense set of unique real and virtual integers, it could not in general distribute itself evenly. That is to say that early phenomena of the universe were lumpy, corresponding more closely to our current calculations than to our current experiments.
Saturday, March 22, 2014