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#1 Re: This is Cool » Prime Numbers!! » 2016-11-15 06:56:51

Hi All

I am switching to this thread because I think this tread is more appropriate for posts dealing with primes. I have previously posted on the "help me" thread.

I am a small time prime hobbyist and have previously posted two "algorithms" that I have developed roughly on my own devices, although I have learned of similar formulations by other researchers/mathematicians, so my stuff might be "copies" or inferior models of what have already been done. I will post my first two "algorithms posted elsewhere on this site for information and comment/criticism!
#1
The 1st algorithm is a variation on the Sieve of Eratosthenes, but actually not a real sieve, but rather a combination of sieve and modulo operations. I discovered recently that there are similar sieves out there (e.g by David Turner of St Andrews University and others -See Wikipedia) so not too sure about the uniqueness of this algorithm therefore. However I did develop the sieve on my own, and I suspect it would be slightly different (and perhaps inferior) to the others. It involves the following; (#1) Take number line of positive integers, 1,2,3....... (#2) strike out all multiples of 2 leaving odd numbers (#3) Identify/mark "3" as a (new) prime number since "3" is not divisible by any positive integer (excluding "1") smaller than itself without leaving a remainder. Calculate square of the newly identified prime (9 in this case). For all odd numbers between 3 and 9 divide by the first prime "2" -all primes less that square root of 9,....nothing to eliminate since 4, 6 & 8 were already eliminated, leaving 5 & 7 as new primes. Now take next prime namely "5". Square of this prime is "25" Now divide all the (odd) numbers remaining on the number line between 9 & 25, by all the identified primes less than "5", namely "2" & "3" . "2" can actually be dropped since all the even numbers had already been eliminated from the number line. This operation eliminates "15,18 & 21", identifying 11,13,17,19 & 23 as primes. (#4) Now do the same for the odd numbers between "25" and the next prime (7=49), dividing with the identified primes less than the square root of (49=7), repeating this operation step by step. The beauty of this operation/algorith is that it identifies primes well ahead of the primes being used in the "sector" being appraised.

#2
My second algorithm is much more curious. We all know that the seemingly random distribution of primes have been a source of amazement to mathematicians for centuries, making it impossible to predict accurately a pattern or the next prime number. However, I have discovered that adding primes together in a particular way (in fact the summing operation of primes seem to produce many solid patterns)-namely consecutive sum of primes produces very REGULAR curves (with coefficients of determination(R^2) of "1" or close to "1"), using the regularity of the obtained curve to "predict" the next prime. This I can do to an accuracy which appear to be an improvement of the estimation provided by the Prime Number Theorum (PNT), and the curve functions anywhere in a series of prime numbers as opposed to the PNT which is more accurate as it approaches infinity. One of the summations is like this (consecutive sum); Add the first two together. Then to the sum add the next prime. To this sum add the next prime number and so on....(p+p), (p+p+p),(p+p+p+p).....This produces a curve of almost perfect slope, whereby it is possible to the predict the NEXT prime by using a trend line or the polynomial equation of the formula. If this curve is true one could potentially predict primes or the location of primes to great numbers. Although I tested the series of consecutive primes in the "Series Encylopedia(OEIS)" it exists, but graphing this series to determine prime numbers has not been done before I think.

I would appreciate further comment.

#2 Re: Help Me ! » More or less Prime » 2016-10-24 07:38:20

Cipolla formula produces 1217 & 1224 with big O (if my calc's are correct). The actual value for 241th prime is 1523.

The polynomial curve (graphic algorithm) is giving +-1500 (vs 1523) by eye-ball.

This is very encouraging.

I am going to try to formalize the algorithm a bit more.

#3 Re: Help Me ! » More or less Prime » 2016-10-19 08:41:04

zetafunc wrote:

There are also other formulae for the nth prime, outside of PNT. For instance, Cipolla showed in 1902 that:

and this has since been improved by other authors, e.g. Dusart.

In the formula do I read log n -1 or log (n-1)? For the big O notation term...do I ignore this term for relatively small numbers?

#4 Re: Help Me ! » More or less Prime » 2016-10-19 07:52:21

What does this formala produce at the 240th prime?

I shall give it a bash, but I think that you might be able to calculate quicker.

#5 Re: Help Me ! » More or less Prime » 2016-10-19 07:49:41

The PNT theorem would be surely the standard to compare any other formula for the prediction of prime numbers against?

So it would be critical to calculate what the PNT ( n log n) would give at given points along the prime number line against any competing recipe.

#6 Re: Help Me ! » More or less Prime » 2016-10-19 07:31:50

What does the equivalent PNT formula give at this point? It was shown before in this tread that the PNT is more inaccurate in this range.

The new predicted number is not far away from the real value...in fact much closer than you would get by pure guessing!!!!

I get 168102 for the actual next term in the series, difference of only 57 for predicted term= 0.338098345% That seems a good forecast?

I am getting a bit electrified!

#7 Re: Help Me ! » More or less Prime » 2016-10-19 06:57:25

PREDICTED PRIME 240th TERM .............. 1531 vs 1511(actual)

The primes around (1511) is 1493,1499---1523,1531


EUREKA??????????????????????

#8 Re: Help Me ! » More or less Prime » 2016-10-19 06:32:16

Thanx!

That's not to bad! for 6th order...that is an error of 0.041428752% for 240th term!!!

What about the 241th term!!!!!!!!!!!!!!!!!!!!

Does fame and fortune beckon us?

#9 Re: Help Me ! » More or less Prime » 2016-10-19 02:42:46

Any body has any information about this....an improved algorithm using Sieve of Eratosthenes by Professor Helfgott (a Peruvian mathematician from France's National Centre for Scientific Research and the University of Göttingen in Germany? See link below

http://www.sciencealert.com/images/articles/processed/SieveofEratosthenes_web_1024.jpg

#10 Re: Help Me ! » More or less Prime » 2016-10-19 02:19:53

Yes that would be good if you could.

I am a bit perplexed however.

If I try to analyse your polynomial evaluations and I test the information with respect to validity, I come up with the following situation

For the set of data that you have analysed (239 terms) the polynomial holds 100% true....now reduce that subset by 1...this new sub set, call it subset b, should still holds true since it is part of a true set a.

Now if we take this subset b we should be able to prove that it holds true for nth term +1, since we know that that is the case...this would seem to indicate that the set is indeed polynomic unless the terms in the original "confined" set is not truely polynomial in the first place?

Why would Excell return a R^2 of 1.0000 for the 6th order polynomial of the curve of the data? Am I missing something, maybe a hidden force pattern in any summation of data. However, if I try to change any term by small units, then the R^2 value reacts very quickly.

Rgds

#11 Re: Help Me ! » More or less Prime » 2016-10-18 23:20:33

Hi Bobbym

Not sure where we are missing each other. When I generate an Excel curve with this data (up to 239th term) I get a trend line equation of
y=3.2759 x^2 - 116.67 x + 1763.3 (R^2=0.999)

Predicting the 240th prime I get a sum of 161002 (vs actual of 165048)...error of 2.45%, giving predicted prime near 1452 vs actual 1511, giving a location error of 4.2%


Prime Number Theory (PNT) -240th prime

~n log n gives prime value of 1315 vs actual of 1511 and location ERROR of 16.7%. (Correction factor not considered)


My problem is that the Excel graph equation for the 6th-order polynomial does not give the expected improved accuracy compared to the 2nd order polynomial, although the coefficient of determination (R^2) has improved to 1.00000

#12 Re: Help Me ! » More or less Prime » 2016-10-18 16:45:39

Thanx!

Could you just have a look a bit higher up the curve, as the slope at the beginning of the curve is at its steepest...remember even PNT has this problem of becoming more accurate only at infinity.

Is that possible?

#13 Re: Help Me ! » More or less Prime » 2016-10-18 08:27:07

Correct....the best fit you can do, so that we could test the predictive strength of such a curve/equation to predict the location and value of sums of primes and therefore the primes as well.

I need an equation for the first set of data listed.

#14 Re: Help Me ! » More or less Prime » 2016-10-18 08:13:22

Please find an equation for the resultant curve to the highest degree of accuracy that you can within your means.

Thx

#15 Re: Euler Avenue » Was mathematics invented or discovered? » 2016-10-18 07:49:31

Was mathematics invented or discovered....an application or a principle. I think a bit of both. Math only exists within consciousness...no consciousness no math! and yes other animals do have degrees of consciousness as well and some perhaps much more.

The problem with mathematics is that it is inherently contradictory/paradoxical.

Consider this, before we can think about something like e.g a math problem....we need time, say 1 second. But 1 second is composed of an infinite amount of fractions of a second. So before we can pass the threshold of 1 second, we first have to pass all these smaller fractions of time....this is an impossibility due to the infinite amount of fractions of time between the zero moment and 1 second...so no thought is possible mathematically.

In fact life is not possible...since there was no time to create it.

#16 Re: Euler Avenue » My theory of the Universe » 2016-10-18 07:29:18

Interesting.

Einstein's Theory of Relativity includes a similar concept...curved space. It is postulated that in curved space you could throw a dart in one direction into space... and it would ultimately return from the opposite direction, after an infinite length of time.

How do you think the universe (energy & matter) came into being w.r.t the theory of the conservation of energy & matter?

Rgds

#17 Re: Help Me ! » surds » 2016-10-18 06:50:30

Substituting back in would give

2[√5-√(2x3)]/2 = √5 - √6

(Please check all answers for accuracy before accepting smile )

#18 Re: Help Me ! » More or less Prime » 2016-10-18 06:36:43

2    5    10    17    28    41    58    77    100    129    160    197    238    281    328    381    440    501    568    639    712    791    874    963    1060    1161    1264    1371    1480    1593    1720    1851    1988    2127    2276    2427    2584    2747    2914    3087    3266    3447    3638    3831    4028    4227    4438    4661    4888    5117    5350    5589    5830    6081    6338    6601    6870    7141    7418    7699    7982    8275    8582    8893    9206    9523    9854    10191    10538    10887    11240    11599    11966    12339    12718    13101    13490    13887    14288    14697    15116    15537    15968    16401    16840    17283    17732    18189    18650    19113    19580    20067    20554    21045    21544    22047    22556    23077    23600    24141    24688    25245    25808    26377    26948    27525    28112    28705    29304    29905    30512    31125    31742    32361    32992    33633    34276    34923    35576    36235    36896    37569    38246    38929    39620    40321    41030    41749    42476    43209    43948    44691    45442    46199    46960    47729    48502    49289    50086    50895    51706    52527    53350    54177    55006    55845    56698    57555    58414    59277    60154    61035    61918    62805    63712    64623    65542    66471    67408    68349    69296    70249    71216    72187    73164    74147    75138    76135    77144    78157    79176    80197    81228    82261    83300    84349    85400    86461    87524    88593    89680    90771    91864    92961    94064    95173    96290    97413    98542    99693    100846    102009    103180    104361    105548    106741    107942    109155    110372    111595    112824    114055    115292    116541    117800    119077    120356    121639    122928    124219    125516    126817    128120    129427    130746    132067    133394    134755    136122    137495    138876    140275    141684    143107    144534    145963    147396    148835    150282    151733    153186    154645    156116    157597    159080    160567    162056    163549    165048    166559

(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    487    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)

The first set is the result of the consecutive sum of the first 240 prime numbers, and the second set in brackets, is the first 240 prime numbers for reference only

I used n=1000 (PNT) which yields a result of 6908

for n log n. The actual value for the 1000th  prime is 7919

#19 Re: Help Me ! » surds » 2016-10-18 05:29:33

Let sqrt of 2 = a & sqrt of 3 = b and sqrt of 5 =c, the equation becomes
>> [(1+a)/(c+b)]+[(1-a)/(c-b)]
>> {[(1+a)(c-b)]+[(1-a)(c+b)]}/(c+b)(c-b)       ....LCD
>> [(c-b+ac-ab)+(c+b-ac-ab)]/(c^2-b^2)         ....simplification
>> 2(c-ab)/(c^2-b^2)                                      ....final simplification

From there you can substitute the values for a,b & c.

Please forgive my notation (answers always to be checked/confirmed!)

Rgds

#20 Re: Help Me ! » More or less Prime » 2016-10-17 10:28:37

Yes please!....However it seems that the PNT is poorly accurate in certain ranges as well?

Should I post a larger data set for analysis?

#21 Re: Help Me ! » More or less Prime » 2016-10-17 08:48:41

Plotting the limited set of listed data only in post #7, generates a curve with an equation for the trend-line (to the 2nd degree of the polynomial) of

y=1.9294 x^2 -7.7876 x + 14.478 (R^2=0.9997)

Substituting for the 5th term give a value of 24 (actual 28) and the 10th term 130 (actual 129). The equation could be refined to the 6th level polynomial in Excel which should theoretically give greater accuracies -However, this does not always seem to be the case due to the % error of the displayed Excel polynomial equations.

The extrapolated "next term-20th term" (following 568 (19th term in the data set) is 630 vs actual of 639 (20th term in prime number set)...the next prime would as per the Excel curve would then be 630-563= 67 vs actual 71(20th prime)...[The 19th prime is 67!!!]

#22 Re: Help Me ! » More or less Prime » 2016-10-17 08:04:25

Hi Zetafunc

How accurate is the PNT at lower values? And how do I use any of these two formulas to say determine the 10th prime which is 29?

#23 Re: Help Me ! » More or less Prime » 2016-10-17 07:10:29

2nd "algorithm" for possible determining of prime numbers.

We are all aware of the notorious "irregularity" of prime numbers w.r.t the gaps between prime numbers and consequently the seemingly pointless exercise of trying to generate a trend/prediction line for the set of prime numbers.

However, the curve for the consecutive sum of prime numbers (and some other derivations of the sum of the list of prime numbers) appear to generate a curve that gives a R^2 (Coefficient of determination) of 1 (correct to at six significant following  zeros as per excel result.

So we start with 2 (1st term), add the next prime ~2+3 (2nd term), then add the next prime to the 2nd term to get the 3rd term, ect....this data seem to generate a Curve that readily delivers a R^2 of 1.00000. I have then used the Equation of the Curve generated by Excel to determine the next prime! or relatively close to the real prime value using this Curve.
However, the problem with the excel generated equation is that even at the 6th level polynomial, the equation does not hold true for the set of data plotted due to the error factor of the displayed Excel equation for the Curve - I have a very limited capacity laptop.

I am wondering 1) if anybody could comment on the potential of this approach towards the prediction of the "next prime" (or the location of a particular prime in the data set of primes plotted. 2) if somebody that has access to more powerful computing power could generate a more accurate Equation for the Curve, which could then be tested to see how far the resultant equation could be stretched to produce accurate values and location of prime numbers in the prime number set. The initial data for the mentioned realtionship is as follows....2,5,10,17,28,41,58,77,100,129,160,197,238,281,328,381,440,501,568.....

Any comments appreciated

#24 Re: Help Me ! » More or less Prime » 2016-10-16 17:03:41

I have developed the algorithm by myself, although I do not want to claim that this algorithm is unique at all and is very similar to the Sieve of Eratosthenes and others I suppose. I see the value of the Algorithm as possibly enabling the determination of a set of prime numbers within a narrow band or range of two (squared) consecutive prime numbers.

Appreciate your comments....very positive since I am only an amateur. I am off to work so I will post the graphic algorithm later today.

Rgds

#25 Re: Help Me ! » More or less Prime » 2016-10-16 08:43:35

I shall indeed be cautious and respectful.

My first algorithm is like this:

Starting off with the counting number line (excluding 1) all the multiples of 2 are eliminated (by sieving), leaving "2" and all the positive odd numbers. "2" is marked as a prime number, not being divisible by any smaller prime number.
Next all the odd numbers between the 2 and the square of (3)=9 are listed, which would be 3,5 & 7. These numbers are divided by the marked prime 2 using Excel Modulo function. If the operation yields a remainder, then this number is marked as a prime number, which would be the case for 3,5 & 7. Now 2,3,5 & 7 would be marked/identified as prime numbers.
Next all the remaining odd numbers between the square of (3)=9 and the square of (5)=25 are highlighted, namely 11,13,15,17,19,21 & 23. These highlighted odd numbers are now divided by the prior "identified" prime numbers - 2 & 3 less than the square root of (25)=5. If these highlighted numbers returns remainders for both the division operations for that number, that number is marked as a prime number. (If any division by the "prior" prime numbers should return a null remainder, then that number would not be prime). This would return 11,13,17,19 & 23 as prime numbers following division by 2 & 3, all these numbers returning remainders for both divisions by 2 & 3.
Next the odd numbers between (5) squared=25 and the square of (7)=49...the next identified prime number are highlighted, namely 29,31,37,41,43 & 47. These numbers are now tested by division by the three prior identified prime numbers (less than the square root of 49) -2,3 & 5 in the same manner....this operation would return 29,31,37,41 & 43 as the prime numbers.
Next the odd numbers between the square of (7)=49 and the next identified prime(11)=121 would be highlighted, divided by the next group of identified primes less than the square root of the highest prime square being tested, namely 2,3,5 and 7, following the same procedure/operations.

In this way primes would be identified over smaller "consecutive" ranges, constantly building up the list between consecutive squares of identified primes, as high as one would like to go.

Comments would be appreciated

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