Math Is Fun Forum

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?

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.

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

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

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

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

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

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.

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.

Substituting back in would give

2[√5-√(2x3)]/2 = √5 - √6

(Please check all answers for accuracy before accepting )

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

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!!!]

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

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