Consecutive cumulative prime sums

[gophne]

Hi Apologies if i might be wasting your time...I am new to the Mersenne forum. I have come across a phenomenon whereby the graph of the consecutive "cummulative sums of prime numbers" generate very regular polynomial graphs for a wide range of prime numbers (from any starting point), e.g. 2,(2+3),(2+3+5),(2+3+5+7),(2+3+5+7+11)(2+3+5+7+11+13).....nth term produces a trendline which generate a coefficient of determination (R^2)~1, thereby making it possible using the trendline to fairly accurately estimate the "next" prime (using the series formula). Could this phenomenon possibly be used for very high prime numbers as well. I have only done work in excel, with the limitations that that imposes w.r.t the magnitudes of prime numbers possible, but the polynomial nature of the grahs have always tended to produce very smooth curves with a CoD~1

[gophne]

Hi

I have come across the following phenomenon with respect to the consecutive sum of cumulative primes - whereby the graphs of the consecutive "cumulative sums of prime numbers" generate very regular polynomial graphs for a wide range of prime numbers (from any prime starting point of any length), e.g. {2,(2+3),(2+3+5),(2+3+5+7),(2+3+5+7+11),(2+3+5+7+11+13).........nth term}, or {53,59,61,67,71,73,79,83,89,97,101} produces a trendline which generate a coefficient of determination (R^2)~1, thereby making it possible using the trendline to fairly accurately estimate the "next" prime (using the series formula). Could this phenomenon possibly be used for very high prime ranges as well. I have only worked in excel, with the limitations that that imposes w.r.t the magnitudes of prime numbers possible, but the polynomial nature of the grahs have always tended to produce very smooth curves with a CoD~1

I have used this relationship/curve to predict the "next" prime fairly accurately within the magnitudes permitted by excel spreadsheets, using the known prime numbers as a template.

I would think that if accurate enough, this relationship might assist to narrow down the search areas in the higher prime ranges if found to hold at the very high prime ranges as well.

[Batalov]

If you take a thousand people, measure their heights and plot a "cumulative sums of heights" you will get a R² of 1.0000 (even after the first hundred), but that doesn't mean that you will be able to "predict" the heights of the 101st, 102nd, etc person rounded to a millimeter or even rounded to a foot.

Same thing in your summation exercise. You will be able to "predict" the "cumulative sums of prime numbers" with good relative precision and CoD, yes, sure. (But you can do it without summing them up just as well, because of what is already known about prime numbers.)

[CRGreathouse]

Well, let's try it out!

The sum of the first 4194304 primes is 144991244981985. What is the 4194305th prime?

[CRGreathouse]

Or better yet (randomly generated):

The sum of the first 3117913001 primes is 114075596703327364352. What is the 3117913002nd prime?

[R. Gerbicz]

Originally Posted by CRGreathouse

Or better yet (randomly generated):

The sum of the first 3117913001 primes is 114075596703327364352. What is the 3117913002nd prime?

Here:

f(s,n)=2*s/n+n/2

is a good approx. for the n-th prime (or for the (n+1)-th prime, there is a little difference between them), in your case this suggests: 74733286039 for the n-th prime.

[a1call]

Sum of consecutive (or random but distinct, for that matter) primes has the following definitive characteristic which may or may not be related to this thread.
Odd addends produces highly undivisible sums, while even sums produce highly composite sums.

[Batalov]

Originally Posted by a1call

produces highly undivisible sums,

Highly compared to what?

Originally Posted by a1call

while even sums produce highly composite sums.

Highly compared to what?

Define 'highly'.

[a1call]

Originally Posted by Batalov

Highly compared to what?

Highly compared to what?

Define 'highly'.

To exaggerate the principle for clarity, the following code starts with a primorial seed (highly composite to start with) rather than 0 and demonstrates, dominance of higher number of prime-factors (not to mention factors) for even sums, than odd sums (not counting the seed).

print("\nBML-100-A-.gp\n")

s=2*3*5*7*11
flag=0
forprime(p=13,367,{
  flag=flag+1;
  if(flag/2!=flag\2,print("\n"));
  s=s+p;
  print(s," * ",factor(s));
})

Result:

BML-100-A-.gp

2310
0


2323 * [23, 1; 101, 1]
2340 * [2, 2; 3, 2; 5, 1; 13, 1]


2359 * [7, 1; 337, 1]
2382 * [2, 1; 3, 1; 397, 1]


2411 * Mat([2411, 1])
2442 * [2, 1; 3, 1; 11, 1; 37, 1]


2479 * [37, 1; 67, 1]
2520 * [2, 3; 3, 2; 5, 1; 7, 1]


2563 * [11, 1; 233, 1]
2610 * [2, 1; 3, 2; 5, 1; 29, 1]


2663 * Mat([2663, 1])
2722 * [2, 1; 1361, 1]


2783 * [11, 2; 23, 1]
2850 * [2, 1; 3, 1; 5, 2; 19, 1]


2921 * [23, 1; 127, 1]
2994 * [2, 1; 3, 1; 499, 1]


3073 * [7, 1; 439, 1]
3156 * [2, 2; 3, 1; 263, 1]


3245 * [5, 1; 11, 1; 59, 1]
3342 * [2, 1; 3, 1; 557, 1]


3443 * [11, 1; 313, 1]
3546 * [2, 1; 3, 2; 197, 1]


3653 * [13, 1; 281, 1]
3762 * [2, 1; 3, 2; 11, 1; 19, 1]


3875 * [5, 3; 31, 1]
4002 * [2, 1; 3, 1; 23, 1; 29, 1]


4133 * Mat([4133, 1])
4270 * [2, 1; 5, 1; 7, 1; 61, 1]


4409 * Mat([4409, 1])
4558 * [2, 1; 43, 1; 53, 1]


4709 * [17, 1; 277, 1]
4866 * [2, 1; 3, 1; 811, 1]


5029 * [47, 1; 107, 1]
5196 * [2, 2; 3, 1; 433, 1]


5369 * [7, 1; 13, 1; 59, 1]
5548 * [2, 2; 19, 1; 73, 1]


5729 * [17, 1; 337, 1]
5920 * [2, 5; 5, 1; 37, 1]


6113 * Mat([6113, 1])
6310 * [2, 1; 5, 1; 631, 1]


6509 * [23, 1; 283, 1]
6720 * [2, 6; 3, 1; 5, 1; 7, 1]


6943 * [53, 1; 131, 1]
7170 * [2, 1; 3, 1; 5, 1; 239, 1]


7399 * [7, 2; 151, 1]
7632 * [2, 4; 3, 2; 53, 1]


7871 * [17, 1; 463, 1]
8112 * [2, 4; 3, 1; 13, 2]


8363 * Mat([8363, 1])
8620 * [2, 2; 5, 1; 431, 1]


8883 * [3, 3; 7, 1; 47, 1]
9152 * [2, 6; 11, 1; 13, 1]


9423 * [3, 3; 349, 1]
9700 * [2, 2; 5, 2; 97, 1]


9981 * [3, 2; 1109, 1]
10264 * [2, 3; 1283, 1]


10557 * [3, 3; 17, 1; 23, 1]
10864 * [2, 4; 7, 1; 97, 1]


11175 * [3, 1; 5, 2; 149, 1]
11488 * [2, 5; 359, 1]


11805 * [3, 1; 5, 1; 787, 1]
12136 * [2, 3; 37, 1; 41, 1]


12473 * Mat([12473, 1])
12820 * [2, 2; 5, 1; 641, 1]


13169 * [13, 1; 1013, 1]
13522 * [2, 1; 6761, 1]


13881 * [3, 1; 7, 1; 661, 1]
14248 * [2, 3; 13, 1; 137, 1]

[CRGreathouse]

Originally Posted by R. Gerbicz

Here:

f(s,n)=2*s/n+n/2

is a good approx. for the n-th prime (or for the (n+1)-th prime, there is a little difference between them), in your case this suggests: 74733286039 for the n-th prime.

Yes, that's reasonable. A different approach would be to throw away the sum and simply invert the logarithmic integral at n.

But in any case the real question is not what you nor I can produce but what gophne can.

[gophne]

Hi

Thanx for the replies.

@Batalov....taking a set of random numbers or even artificially jerking the values does not produce a smooth curve or R^2 ~ 1.00000

@CRGreathouse...the relationship is not of the sum of consecutive primes per se, but for the sum of consecutive CUMULATIVE SUB-SUMS if you like. From the resultant (equation of the) polynomial trendline it is then possible to estimate/predict the next value in the series as per trend-line definition.

My question is how does this prediction (curve trendline) compares to the standard PNT approximations. My experience for the lower ranges (excel spreadsheet ranges of prime numbers) is that this predicted values are closer than the PNT approximations which tend to become more accurate at very large values.

If there is any advantage at all, would it be possible to use this "relationship" to narrow the search in the very high prime ranges wherein the more serious prime number researchers are working as well?

Thanx for the other replies...could anybody do a simple comparison for a easily testable range of primes/consecutive cumulative prime "sub-sums" using the conventional PNT-formulas and "trend-line approach?

Thanks so much