Dear Thierry,
I’ll have a look on the prime number theorem, because I think it also implies
on primegaps and how they are distributed,
And those findings on my result were somehow diffirent. For example 10^1 or
10^x.
Mine are distributed somehow diffirent and here’s that results.
In the second line you see how the primenumbers in between are growing. The
gaps are also growing with the formula. What I think is sort of exponential.
Maybe I can’t read it to well to see some resemblance. Those prime ”pakkeges”
are growing nicely,
And I’ll try to create a formula in a mathematical way,
And to find out if there’s some resemblance and so if there maybe is a
diffirence..
They seem to ditribute and grow evenly.
This is made in C++ by me
Results.
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 %
100: 610 247500 <=> 255025 6.1 per gap=>
8.10631 % total=> 8.09448 %
101: 605 255025 <=> 262701 5.9901 per gap=>
7.88171 % total=> 8.10411 %
102: 640 262701 <=> 270530 6.27451 per gap=>
8.17526 % total=> 7.88456 %
103: 632 270530 <=> 278512 6.13592 per gap=>
7.91732 % total=> 8.17278 %
104: 649 278512 <=> 286650 6.24038 per gap=>
7.97493 % total=> 7.91787 %
105: 642 286650 <=> 294945 6.11429 per gap=>
7.7396 % total=> 7.97271 %
106: 663 294945 <=> 303398 6.25472 per gap=>
7.84291 % total=> 7.74057 %
107: 677 303398 <=> 312012 6.3271 per gap=>
7.85976 % total=> 7.84306 %
108: 723 312012 <=> 320787 6.69444 per gap=>
8.23932 % total=> 7.86324 %
109: 712 320787 <=> 329725 6.53211 per gap=>
7.96599 % total=> 8.23683 %
110: 720 329725 <=> 338828 6.54545 per gap=>
7.90991 % total=> 7.96548 %
111: 722 338828 <=> 348096 6.5045 per gap=>
7.78983 % total=> 7.90884 %
112: 734 348096 <=> 357532 6.55357 per gap=>
7.77872 % total=> 7.78973 %
113: 755 357532 <=> 367137 6.68142 per gap=>
7.86049 % total=> 7.77944 %
114: 747 367137 <=> 376912 6.55263 per gap=>
7.64155 % total=> 7.85859 %
115: 768 376912 <=> 386860 6.67826 per gap=>
7.72053 % total=> 7.64223 %
116: 790 386860 <=> 396981 6.81034 per gap=>
7.80555 % total=> 7.72126 %
117: 772 396981 <=> 407277 6.59829 per gap=>
7.49806 % total=> 7.80295 %
118: 818 407277 <=> 417750 6.9322 per gap=>
7.81093 % total=> 7.50069 %
119: 821 417750 <=> 428400 6.89916 per gap=>
7.70856 % total=> 7.81008 %
120: 848 428400 <=> 439230 7.06667 per gap=>
7.8301 % total=> 7.70956 %
121: 849 439230 <=> 450241 7.01653 per gap=>
7.71047 % total=> 7.82912 %
122: 840 450241 <=> 461434 6.88525 per gap=>
7.50436 % total=> 7.7088 %
123: 880 461434 <=> 472812 7.15447 per gap=>
7.73456 % total=> 7.50621 %
124: 898 472812 <=> 484375 7.24194 per gap=>
7.76615 % total=> 7.73482 %
125: 898 484375 <=> 496125 7.184 per gap=>
7.64255 % total=> 7.76517 %
126: 901 496125 <=> 508064 7.15079 per gap=>
7.54701 % total=> 7.6418 %
127: 928 508064 <=> 520192 7.30709 per gap=>
7.6514 % total=> 7.54783 %
128: 940 520192 <=> 532512 7.34375 per gap=>
7.62987 % total=> 7.65123 %
129: 937 532512 <=> 545025 7.26357 per gap=>
7.48821 % total=> 7.62878 %
130: 952 545025 <=> 557732 7.32308 per gap=>
7.49164 % total=> 7.48824 %
131: 971 557732 <=> 570636 7.41221 per gap=>
7.52509 % total=> 7.49189 %
132: 999 570636 <=> 583737 7.56818 per gap=>
7.62537 % total=> 7.52584 %
133: 1009 583737 <=> 597037 7.58647 per gap=>
7.58647 % total=> 7.62508 %
With friendly regards,
Dirk-Anton Broersen
Verzonden vanuit Mail<https://go.microsoft.com/fwlink/?LinkId=550986> voor
Windows 10
Van: Thierry Arnoux<mailto:[email protected]>
Verzonden: dinsdag 24 maart 2020 15:21
Aan: [email protected]<mailto:[email protected]>
CC: Dirk-Anton Broersen<mailto:[email protected]>
Onderwerp: Re: Prime numbers in Triangular intervals
Hi Dirk-Anton,
This list is more for formalization of mathematics in Metamath; other mailing
lists are probably more adequate, like alt.math.undergrad
(http://mathforum.org/library/view/6791.html
<http://mathforum.org/library/view/6791.html> ).
Anyway, if you are interested in how prime numbers are distributed, you should
check the prime number theorem
(https://en.wikipedia.org/wiki/Prime_number_theorem?wprov=sfti1).
Furthermore, here is what I would suggest:
You may be able to write your sequence in a closed form. You define it as
A_n=(T_n)^2/n, where Tn is the nth triangular number. There is a closed form
for the triangular numbers: Tn = n(n+1)/2, and if you inject it in A_n, you get
A_n = n(n+1)^2/4.
Then, using the prime number theorem, you may be able to estimate how many
primes are lower than or equal to A_n; how many are less than A_(n+1), and by
difference, how many primes are between A_n and A_(n+1), and finally, you could
check if this agrees with the limit you get experimentally by counting.
BR,
_
Thierry
Envoyé de mon iPhone
Van: Dirk-Anton Broersen<mailto:[email protected]>
Verzonden: dinsdag 24 maart 2020 13:08
Aan: [email protected]<mailto:[email protected]>; Dirk-Anton
Broersen<mailto:[email protected]>
Onderwerp: Re: [Metamath] Formalizing IMO B2.1972
I'm also a beginner. And I received this email. I posted lately an email about
a finding. I don 't know of it's unique or known or if it has resemblance.
It's also about triangelar numbers in a formula.
E
x = x + 1
(triangelar number) power 2 / x
triangelar number = triangelar number + triangelar number + 1
First results are and I also wrote a programm in c++ wich you can copy paste to
cpp.sh to see the results.
1 1 (1/1) 1 = 1 ^2
2 4.5 (9/2) 9 = 3 ^2
3 12 (36/3) 36 = 6 ^2
4 25 (100/4) 100 = 10 ^2
1 <==> 4.5 <==> 12 <==> 25 <==> ..
within these gaps there is an amount of primenumbers that inscrease. Percentual
it's also intersting.
I'll send next the first number of results of the programm. then it's also
clear what number of primes are increasing.
Including the programm.
I don 't wanna frustrate others work. This might be seen as trolling. I just
received this email, but I tought this might be something. I'm an
undergraduated mathematician. And it has also to do with triangelar numbers.
With friendly regards,
Dirk-Anton Broersen
Outlook for Android<https://aka.ms/ghei36> downloaden
<77609AC6AB764EF7881D5C907B5BE9D9.png>
From: 'Stanislas Polu' via Metamath <[email protected]>
Sent: Monday, March 23, 2020 9:05:17 PM
To: [email protected] <[email protected]>
Subject: Re: [Metamath] Formalizing IMO B2.1972
Hi Marnix!
Thanks for sharing. The proof I formalized[0] is very closed but I agree is
also a bit more complicated.
Out of curiosity, where did you find that proof which has a very "formal"
presentation?
Best,
-stan
[0] http://us.metamath.org/mpeuni/imo72b2.html
On Mon, Mar 23, 2020 at 6:38 PM Marnix Klooster
<[email protected]<mailto:[email protected]>> wrote:
Hi Stan,
If I were to formalize this in Metamath, I'd use the first proof, but in a more
calculational format.
I've attached it, unfortunately as a picture.
Yes, this is a longer proof, but it seems somehow easier to me.
Hope this helps someone... :-)
<image.png>
Groetjes,
<><
Marnix
Op do 27 feb. 2020 om 18:08 schreef 'Stanislas Polu' via Metamath
<[email protected]<mailto:[email protected]>>:
Hi all,
I'm quite a beginner with Metamath (I have read a bunch of proofs, most of the
metamath book, I have implemented my own verifier, but I haven't constructed
any original proof yet) and I am looking to formalize the following proof:
IMO B2 1972: http://www.cs.ru.nl/~freek/demos/exercise/exercise.pdf
Alternative version: http://www.cs.ru.nl/~freek/demos/exercise/exercise2.pdf
(More broadly, I think this would be an interesting formalization to have in
set.mm<http://set.mm> given this old but nonetheless interesting page:
http://www.cs.ru.nl/~freek/demos/index.html)
I am reaching out to the community to get direction on how should I go about
creating an efficient curriculum for myself in order to achieve that goal? Any
other advice is obviously welcome!
Thank you!
-stan
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