User:Proofname123/Sandbox/Primenumber growth within exponential gaps
Some reseach I did on wikipedia and the next findings can resemble some other conjectures or mathematical functions.
For instance Chebyshev function. source=> https://en.wikipedia.org/wiki/Chebyshev_function#The_exact_formula
Or the Goldbach conjecture. source=> https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
1. What is the formula about?
2. the formula has twoo components.
3. The formula.
4. The first hundred results.
5. Does this formula already excists?
6. The programm in c++
7. resemblence with other formulas and the meaning of this formula.
1. What is the formula about?
In this formula, what might be seen as a conjecture, some findings resemble an evenly growth of primenumbers within an exponential growth of natural numbers.
2. the formula has twoo components.
The formula has twoo components.
- the growth of natural number gaps expanding exponential - the growth of a quantity of primenumbers within the gaps as ment above
3. The formula.
1²/1 = 1
3²/2 = 4.5
6²/3 = 12
10²/4 = 25
...
the input is 1² as an input of a triangular number so in the second the input is 3. Divided by a consecutive number adding from 1 + 1 etc. So triangular number ²/ natural consecutive number
In between those results, a gap, there is an amount of primenumbers. A substantial growth is seen in the results of the execution of a c++ programm.
4. the first 99 resulst.
These resulst come from a c++ program.
number of gap | amount of primenumbers | the gaps
1: 1 0 <=> 1 1 per gap=> 100 % total=> 50 %
2: 2 1 <=> 4.5 1 per gap=> 57.1429 % total=> 85.7143 %
3: 3 4.5 <=> 12 1 per gap=> 40 % total=> 52.8571 %
4: 4 12 <=> 25 1 per gap=> 30.7692 % total=> 38.1538 %
5: 5 25 <=> 45 1 per gap=> 25 % total=> 29.8077 %
6: 7 45 <=> 73.5 1.16667 per gap=> 24.5614 % total=> 24.9373 %
7: 9 73.5 <=> 112 1.28571 per gap=> 23.3766 % total=> 24.4133 %
8: 8 112 <=> 162 1 per gap=> 16 % total=> 22.557 %
9: 11 162 <=> 225 1.22222 per gap=> 17.4603 % total=> 16.146 %
10: 14 225 <=> 302.5 1.4 per gap=> 18.0645 % total=> 17.5152 %
11: 15 302.5 <=> 396 1.36364 per gap=> 16.0428 % total=> 17.896 %
12: 19 396 <=> 507 1.58333 per gap=> 17.1171 % total=> 16.1254 %
13: 19 507 <=> 637 1.46154 per gap=> 14.6154 % total=> 16.9384 %
14: 23 637 <=> 787.5 1.64286 per gap=> 15.2824 % total=> 14.6599 %
15: 25 787.5 <=> 960 1.66667 per gap=> 14.4928 % total=> 15.233 %
16: 29 960 <=> 1156 1.8125 per gap=> 14.7959 % total=> 14.5106 %
17: 29 1156 <=> 1377 1.70588 per gap=> 13.1222 % total=> 14.7029 %
18: 37 1377 <=> 1624.5 2.05556 per gap=> 14.9495 % total=> 13.2183 %
19: 33 1624.5 <=> 1900 1.73684 per gap=> 11.9782 % total=> 14.8009 %
20: 38 1900 <=> 2205 1.9 per gap=> 12.459 % total=> 12.0011 %
21: 43 2205 <=> 2541 2.04762 per gap=> 12.7976 % total=> 12.4744 %
22: 50 2541 <=> 2909.5 2.27273 per gap=> 13.5685 % total=> 12.8311 %
23: 45 2909.5 <=> 3312 1.95652 per gap=> 11.1801 % total=> 13.469 %
24: 57 3312 <=> 3750 2.375 per gap=> 13.0137 % total=> 11.2535 %
25: 56 3750 <=> 4225 2.24 per gap=> 11.7895 % total=> 12.9666 %
26: 61 4225 <=> 4738.5 2.34615 per gap=> 11.8793 % total=> 11.7928 %
27: 62 4738.5 <=> 5292 2.2963 per gap=> 11.2014 % total=> 11.8551 %
28: 74 5292 <=> 5887 2.64286 per gap=> 12.437 % total=> 11.244 %
29: 68 5887 <=> 6525 2.34483 per gap=> 10.6583 % total=> 12.3777 %
30: 77 6525 <=> 7207.5 2.56667 per gap=> 11.2821 % total=> 10.6784 %
31: 83 7207.5 <=> 7936 2.67742 per gap=> 11.3933 % total=> 11.2855 %
32: 83 7936 <=> 8712 2.59375 per gap=> 10.6959 % total=> 11.3721 %
33: 95 8712 <=> 9537 2.87879 per gap=> 11.5152 % total=> 10.72 %
34: 94 9537 <=> 10412.5 2.76471 per gap=> 10.7367 % total=> 11.4929 %
35: 96 10412.5 <=> 11340 2.74286 per gap=> 10.3504 % total=> 10.726 %
36: 101 11340 <=> 12321 2.80556 per gap=> 10.2956 % total=> 10.3489 %
37: 114 12321 <=> 13357 3.08108 per gap=> 11.0039 % total=> 10.3143 %
38: 110 13357 <=> 14449.5 2.89474 per gap=> 10.0686 % total=> 10.9799 %
39: 124 14449.5 <=> 15600 3.17949 per gap=> 10.7779 % total=> 10.0864 %
40: 121 15600 <=> 16810 3.025 per gap=> 10 % total=> 10.7589 %
41: 133 16810 <=> 18081 3.2439 per gap=> 10.4642 % total=> 10.0111 %
42: 125 18081 <=> 19414.5 2.97619 per gap=> 9.37383 % total=> 10.4388 %
43: 147 19414.5 <=> 20812 3.4186 per gap=> 10.5188 % total=> 9.39985 %
44: 150 20812 <=> 22275 3.40909 per gap=> 10.2529 % total=> 10.5129 %
45: 153 22275 <=> 23805 3.4 per gap=> 10 % total=> 10.2474 %
46: 153 23805 <=> 25403.5 3.32609 per gap=> 9.57147 % total=> 9.99088 %
47: 168 25403.5 <=> 27072 3.57447 per gap=> 10.0689 % total=> 9.58184 %
48: 169 27072 <=> 28812 3.52083 per gap=> 9.71264 % total=> 10.0617 %
49: 165 28812 <=> 30625 3.36735 per gap=> 9.10094 % total=> 9.70041 %
50: 187 30625 <=> 32512.5 3.74 per gap=> 9.90728 % total=> 9.11675 %
51: 193 32512.5 <=> 34476 3.78431 per gap=> 9.82939 % total=> 9.90579 %
52: 188 34476 <=> 36517 3.61538 per gap=> 9.21117 % total=> 9.81772 %
53: 199 36517 <=> 38637 3.75472 per gap=> 9.38679 % total=> 9.21442 %
54: 206 38637 <=> 40837.5 3.81481 per gap=> 9.36151 % total=> 9.38633 %
55: 230 40837.5 <=> 43120 4.18182 per gap=> 10.0767 % total=> 9.37428 %
56: 210 43120 <=> 45486 3.75 per gap=> 8.87574 % total=> 10.0556 %
57: 224 45486 <=> 47937 3.92982 per gap=> 9.13913 % total=> 8.88028 %
58: 239 47937 <=> 50474.5 4.12069 per gap=> 9.41872 % total=> 9.14387 %
59: 239 50474.5 <=> 53100 4.05085 per gap=> 9.10303 % total=> 9.41346 %
60: 246 53100 <=> 55815 4.1 per gap=> 9.06077 % total=> 9.10234 %
61: 269 55815 <=> 58621 4.40984 per gap=> 9.5866 % total=> 9.06925 %
62: 257 58621 <=> 61519.5 4.14516 per gap=> 8.86666 % total=> 9.57517 %
63: 265 61519.5 <=> 64512 4.20635 per gap=> 8.85547 % total=> 8.86648 %
64: 282 64512 <=> 67600 4.40625 per gap=> 9.13212 % total=> 8.85973 %
65: 274 67600 <=> 70785 4.21538 per gap=> 8.60283 % total=> 9.1241 %
66: 297 70785 <=> 74068.5 4.5 per gap=> 9.04523 % total=> 8.60943 %
67: 302 74068.5 <=> 77452 4.50746 per gap=> 8.92567 % total=> 9.04347 %
68: 314 77452 <=> 80937 4.61765 per gap=> 9.01004 % total=> 8.92689 %
69: 319 80937 <=> 84525 4.62319 per gap=> 8.89075 % total=> 9.00834 %
70: 315 84525 <=> 88217.5 4.5 per gap=> 8.53081 % total=> 8.88568 %
71: 333 88217.5 <=> 92016 4.69014 per gap=> 8.76662 % total=> 8.53408 %
72: 355 92016 <=> 95922 4.93056 per gap=> 9.08858 % total=> 8.77103 %
73: 344 95922 <=> 99937 4.71233 per gap=> 8.56787 % total=> 9.08155 %
74: 352 99937 <=> 104062 4.75676 per gap=> 8.5323 % total=> 8.5674 %
75: 364 104062 <=> 108300 4.85333 per gap=> 8.58997 % total=> 8.53306 %
76: 371 108300 <=> 112651 4.88158 per gap=> 8.52678 % total=> 8.58915 %
77: 379 112651 <=> 117117 4.92208 per gap=> 8.48634 % total=> 8.52626 %
78: 400 117117 <=> 121700 5.12821 per gap=> 8.72886 % total=> 8.48941 %
79: 400 121700 <=> 126400 5.06329 per gap=> 8.50973 % total=> 8.72612 %
80: 406 126400 <=> 131220 5.075 per gap=> 8.42324 % total=> 8.50867 %
81: 417 131220 <=> 136161 5.14815 per gap=> 8.43959 % total=> 8.42344 %
82: 438 136161 <=> 141224 5.34146 per gap=> 8.65014 % total=> 8.44212 %
83: 429 141224 <=> 146412 5.16867 per gap=> 8.26988 % total=> 8.64562 %
84: 457 146412 <=> 151725 5.44048 per gap=> 8.60154 % total=> 8.27378 %
85: 447 151725 <=> 157165 5.25882 per gap=> 8.21691 % total=> 8.59707 %
86: 461 157165 <=> 162734 5.36047 per gap=> 8.27871 % total=> 8.21762 %
87: 458 162734 <=> 168432 5.26437 per gap=> 8.0372 % total=> 8.27597 %
88: 489 168432 <=> 174262 5.55682 per gap=> 8.38765 % total=> 8.04114 %
89: 501 174262 <=> 180225 5.62921 per gap=> 8.40181 % total=> 8.38781 %
90: 511 180225 <=> 186322 5.67778 per gap=> 8.38048 % total=> 8.40158 %
91: 505 186322 <=> 192556 5.54945 per gap=> 8.10139 % total=> 8.37745 %
92: 524 192556 <=> 198927 5.69565 per gap=> 8.22477 % total=> 8.10271 %
93: 522 198927 <=> 205437 5.6129 per gap=> 8.01843 % total=> 8.22257 %
94: 562 205437 <=> 212088 5.97872 per gap=> 8.45049 % total=> 8.02298 %
95: 536 212088 <=> 218880 5.64211 per gap=> 7.89106 % total=> 8.44466 %
96: 572 218880 <=> 225816 5.95833 per gap=> 8.24683 % total=> 7.89472 %
97: 579 225816 <=> 232897 5.96907 per gap=> 8.17681 % total=> 8.24611 %
98: 566 232897 <=> 240124 5.77551 per gap=> 7.8312 % total=> 8.17332 %
99: 597 240124 <=> 247500 6.0303 per gap=> 8.09437 % total=> 7.83383 %
You might aswell read the other resulst in percentages wich tells about the growth. I think these results are somewhat spectaculair to see a growth of primenumbers wich sort of evenly grow.
5. Does this formula already excists?
I don't know enough about this and I would like to know. This might resemble other functions or conjectures.
6. The programm in c++ (program language)
#include <iostream> #include <iomanip> #include <cmath>
using namespace std;
int main() {
int count, n, c;
double x = 0; double b = 0; double bnext = 0; double z = 1; double y = 0; double l = 0; double h = 0; double m = 0; double k = 0; double v = 0;
while(x < 10000) {
y = y + z;
z = z + 1.5;
bnext = b;
b = b + y;
l = b *1;
h = bnext *1;
x++;
for (n = h; n <= l; n++){
count = 0;
for (int i = 2; i <= n/2; i++){
if(n%i==0){
count++;
break;
}
}
if(count==0 && n!=1){
c = c + 1;
}
}
k = m;
m = (c/(l-h))*100;
v = ((k*x) +m)/(x+1);
cout << x << ": " << c << " " << h << " <=> " << l << " " << c/x << " per gap=> " << m << " %" << " total=> " << v << " %" << endl;
c= 0;
}
return 0; }
7. resemblance with other formulas and the meaning of this formula.
The question is what this formula says to me and if there are already such formulas that say about thesame thing. To me this formula says that there is an analyse for to track the growth of primenumbers. I'm curious for more results. I came up with it, with considering there are no more numbers than an amount of numbers. Like saying 1, 2, 3, 4 that's all. for instance gap one: here only number one is used (1 x 1 = 1) is also 1² divided by one here only number one and twoo is used (1 x 1 = 1 plus 1 x 2 = 2 plus 2 x 1 = 2 plus 2 x 2 = 4) so 1 + 2 + 2 + 4 = 9 is also 3² divided by twoo etc.