Prime Density
Primes numbers become less frequent as the numbers get larger. For example between 0 and 100 there are 25 primes. From 100 to 200 there are only 21 prime numbers.
If we use this step size of 100 (numbers that end with a double zero), there's a step where there is only one prime number in that range of 100 numbers. Actually there's an infinite number. But what step is the lowest that has only 1 prime?
Sample answer = “1000 to 1100" (that step has 16 primes)
If we use this step size of 100 (numbers that end with a double zero), there's a step where there is only one prime number in that range of 100 numbers. Actually there's an infinite number. But what step is the lowest that has only 1 prime?
Sample answer = “1000 to 1100" (that step has 16 primes)
Labels: mathschallenge
posted by Ragknot at
8:57 AM
29 Comments:
The prime of interest is: 2101249
155600 to 155700: 10
155700 to 155800: 11
155800 to 155900: 11
155900 to 156000: 1
addition to above: 155921 is prime.
Hi EB. You're right. I did a silly and published the 155900th prime. The correct prime of is interest is: 155921. Greetz.
PS I've just fixed a major and a minor howler for the Floating hourglass problem.
kicking my posts through.
Rats! EB beat me to that too, boo hoo ;)
Congrats EB!
Did anyone find a easy way to do this? Or did you just have to spend time browsing?
Chris?
I got an invite from a Chris A.....
Could that be you?
... so I'll overcompensate and give the next 5 primes that that satisfy the problem:
268343,413353,1004873,1140091,1272749.
Ragknot, what is the first prime gap of exactly 100? :) Just for clarity, I mean exactly 100 composite numbers in between. No special boundaries like 100. I've no idea what the answer is - I assume that such a gap exists.
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I did it with Mathematica. It has a lot of number theory functionality built in.
My code (not exactly as I did it originally:
For[m = 150000, m <= 200000, m += 100,
If[PrimePi[m + 100] - PrimePi[m] == 1, Print[{m, NextPrime[m]}]]]
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Ragknot. I haven't invited you to anything.
I realise that my challenge above may be impossible i.e. there may not be such a gap.
Ragknot. How do you handle these potentially difficult prime number problems - is it only BASIC you use, or have you got something more advanced?
Here's the first range that has NO prime numbers at all.
1671800 to 1671900
Ragnot.. I wrote a simple program with a nested loop to find primes. It resets the counter before each 100 block range.
ooh.. just found another range with NO prime numbers.
2637800 to 2637900
Chris,
I hope I understood your query.
The lowest pair of prime numbers seperated by exactly 100 is...
396,733
396,833
Was that your question?
EB. Just in case you didn't know, there are an infinite number of such gaps, of any finite width you care to think of. e.g. there are an infinite number of prime gaps of width > 10000000000000.
The largest gap I observed in consective primes in the first one million primes was 154.
And it occured 3 times.
A prime.........B prime.....Diff
4652507........4652353......154
11114087.......11113933.....154
15204131.......15203977.....154
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Phew, sorted. First 5 gaps of 100 are:
396733, 838249,1313467, 1648081, 1655707
Please agree, PC going into meltdown (not really, I was).
Ragknot, sorry, I deleted a load of posts as they were kick posts and other ramblings. You interpreted the question correctly.
Out of curiosity, have you ever dabbled with unbelievably frustrating Goldbach conjecture?
73 gaps of 100 wide
396833 396733
838349 838249
1313567 1313467
1648181 1648081
1655807 1655707
2346089 2345989
2784473 2784373
3255059 3254959
3595589 3595489
4047257 4047157
4359503 4359403
4571207 4571107
4665653 4665553
4783973 4783873
5211209 5211109
5398697 5398597
5528387 5528287
5723999 5723899
6027383 6027283
6242363 6242263
6429323 6429223
6851963 6851863
7259267 7259167
7554467 7554367
7662617 7662517
8257157 8257057
8350583 8350483
8441969 8441869
8806991 8806891
8841629 8841529
9162467 9162367
9389927 9389827
9413633 9413533
9550913 9550813
9812711 9812611
9918521 9918421
10172321 10172221
10194707 10194607
10295921 10295821
10569107 10569007
10668521 10668421
10670489 10670389
10942853 10942753
10986377 10986277
11367047 11366947
11609303 11609203
11765759 11765659
11936909 11936809
12117221 12117121
12459329 12459229
12468557 12468457
12523073 12522973
12557417 12557317
12623189 12623089
12686747 12686647
13200101 13200001
13251647 13251547
13290647 13290547
13385531 13385431
13660763 13660663
13721033 13720933
13735013 13734913
13776167 13776067
13905707 13905607
14059757 14059657
14076857 14076757
14160977 14160877
14252531 14252431
14272301 14272201
14312381 14312281
14933591 14933491
14986109 14986009
15074099 15073999
The Goldbach conjecture.
I read about it. Two of them actually. Sounded interesting.
Aaah, now I know why you've been so quiet:) Thanks for the confirmation figures (and the rest). Your PC must be getting quite hot by the time you'd got those numbers.
I'd been getting all the prime functions muddled up in Mathematica and getting obviously crazy results.
The other GC! Is that the trivial other one?
Found it. The weak GB with 3 odd primes.
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