mersenneforum.org > Math Primes in millennia
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 2009-02-22, 12:49 #1 mart_r     Dec 2008 you know...around... 28A16 Posts Primes in millennia I had spent some time looking for the successive record minimum number of primes in a "millennium", i.e. intervals between ...000 and ...999. This is how far I've gotten: Code: # of pr. millennium 168 0,___ 135 1,___ 127 2,___ 120 3,___ 119 4,___ 114 5,___ 107 7,___ 106 10,___ 103 11,___ 102 14,___ 98 16,___ 94 18,___ 92 29,___ 90 38,___ 88 40,___ 85 43,___ 80 64,___ 76 88,___ 73 168,___ 71 180,___ 69 212,___ 68 293,___ 67 356,___ 63 452,___ 61 555,___ 59 638,___ 58 871,___ 54 913,___ 53 1,637,___ 52 2,346,___ 46 3,279,___ 43 7,176,___ 42 14,420,___ 38 15,369,___ 36 36,912,___ 35 51,459,___ 34 96,733,___ 33 113,376,___ 31 141,219,___ 28 200,315,___ 27 233,047,___ 26 729,345,___ 25 951,847,___ 24 1,704,275,___ 23 1,917,281,___ 22 2,326,985,___ 21 2,937,877,___ 20 6,973,534,___ 18 7,362,853,___ 17 12,838,437,___ 16 26,480,476,___ 15 34,095,574,___ 13 162,661,473,___ 12 304,552,694,___ 10 378,326,417,___ 9 1,252,542,156,___ 8 3,475,851,270,___ 7 6,603,973,861,___ 6 7,613,200,181,___ I also know that the first millennium without any prime number is 13,893,290,219,204,___ So I thought if I never ask, I may never find out about the gap - the first millennia with 5, 4, 3, 2 primes and one prime respectively. Is someone out there who can provide this information? Last fiddled with by mart_r on 2009-02-22 at 12:50
 2009-02-22, 14:22 #2 m_f_h     Feb 2007 24×33 Posts You should submit this as "records in A038823" with kw "fini". Last fiddled with by m_f_h on 2009-02-22 at 14:23
2009-02-22, 15:12   #3
mart_r

Dec 2008
you know...around...

2·52·13 Posts

Quote:
 Originally Posted by m_f_h You should submit this as "records in A038823" with kw "fini".
I was thinking about it quite some time ago.
Yet 1) there are also some other sequences I still like to submit but I think they are too unimportant * and 2) I'm a bit unsure about what numbers I should choose: 0, 1, 2, 3, 4, 5, 7, 10 ... as the constant digits of the interval or 1, 2, 3, 4, 5, 6, 8, 11 ... as it being the n-th millennium.

* Though there are so many "unimportant" sequences in the database compared to others that aren't in there which I think make it unnecessarily unwieldy.

2009-02-22, 15:24   #4
m_f_h

Feb 2007

24·33 Posts

Quote:
 Originally Posted by mart_r I was thinking about it quite some time ago. Yet 1) there are also some other sequences I still like to submit but I think they are too unimportant * and 2) I'm a bit unsure about what numbers I should choose: 0, 1, 2, 3, 4, 5, 7, 10 ... as the constant digits of the interval or 1, 2, 3, 4, 5, 6, 8, 11 ... as it being the n-th millennium.
That's a good point. Common speak would be in favour of the latter (and btw not 0...999 but 1...1000, but since the 000 is unlikely to yield a prime, there's no problem) - and mathematically speaking there's no difference in saying [m,m+1] or [m-1,m]. Anyway, the essential thing is (IMHO) to "immortalize" the results obtained by the computational effort you invested.

PS1: On a second thought, "record indices of A038823" (in decreasing sense: n such that A038823(n) is lower than all A038823(k) with k<n), would correspond to the former choice ; I think this is preferrable.

Quote:
 Originally Posted by mart_r * Though there are so many "unimportant" sequences in the database compared to others that aren't in there which I think make it unnecessarily unwieldy.
Since by submitting one which is a bit less unimportant than those, you would decrease the "mean unimportance": please go ahead ! :-)

PS: I think the last argument is flawed... :-( ! But don't mind...
Let's say, a nontrivial "records" sequence for an existing sequence with A-number < 50000, which supposedly takes several hours of CPU time (7613200000 - 7613400000 took me about 10 mins) is always worth submitting.

Last fiddled with by m_f_h on 2009-02-22 at 15:39 Reason: several reasons...

2009-02-22, 17:48   #5
CRGreathouse

Aug 2006

10111011000012 Posts

Quote:
 Originally Posted by m_f_h PS1: On a second thought, "record indices of A038823" (in decreasing sense: n such that A038823(n) is lower than all A038823(k) with k
Ditto. I prefer the 0th, 1st, ... 'millennia' counting primes from 1000n to 1000(n+1).

2009-02-22, 19:29   #6
mart_r

Dec 2008
you know...around...

12128 Posts

Quote:
 Originally Posted by m_f_h You should submit this as "records in A038823" with kw "fini".
Done.

I hope all terms I submitted are appearing later. Or do they generally make a cut after three lines?

(by the way - look at this comparison: there are
13 primes between 162,661,473,001 and 162,661,474,000, but
70 primes between 166,007,963,001 and 166,007,964,000 )

2009-02-22, 19:52   #7
CRGreathouse

Aug 2006

32·5·7·19 Posts

Quote:
 Originally Posted by mart_r I hope all terms I submitted are appearing later. Or do they generally make a cut after three lines?
They only display three lines. Other terms can be submitted as a b-file.

Last fiddled with by CRGreathouse on 2009-02-22 at 19:52 Reason: fix spacing

 2009-03-02, 17:50 #8 mart_r     Dec 2008 you know...around... 2·52·13 Posts I'm not very confident about this b-file-thingy. With the sequence itself appearing in the database within 48 hours, I'm now waiting for more than five days for the b-file (which I think is especially important here) to show up. Should I wait another few days or try to re-send it?
 2009-03-02, 18:09 #9 CRGreathouse     Aug 2006 32·5·7·19 Posts Give it two weeks.
 2009-03-04, 19:16 #10 mart_r     Dec 2008 you know...around... 2×52×13 Posts Yep, it's updated now. Another thing I'd like to share is "primes in a quadratic pattern", i.e. p ± n*(n+1) for a run of n's (say up to 12 or 13, starting from 0). I've searched p+n(n+1) to 37*10^12 some time ago and am now with p-n(n+1), currently at ~ 4*10^12. Strangely, the OEIS only lists a sequence for n=0...6, and nothing else. That would require me to send about 18 separate sequences. I was thus thinking about a single sequence for each p+ and p-, listing ten (or more) terms successively for each n. Any opinions/suggestions?
2009-03-04, 19:30   #11
R.D. Silverman

Nov 2003

1D2416 Posts

Quote:
 Originally Posted by mart_r Yep, it's updated now. Another thing I'd like to share is "primes in a quadratic pattern", i.e. p ± n*(n+1) for a run of n's (say up to 12 or 13, starting from 0). I've searched p+n(n+1) to 37*10^12 some time ago and am now with p-n(n+1), currently at ~ 4*10^12. Strangely, the OEIS only lists a sequence for n=0...6, and nothing else. That would require me to send about 18 separate sequences. I was thus thinking about a single sequence for each p+ and p-, listing ten (or more) terms successively for each n. Any opinions/suggestions?
My opinion is that such a thing would be a pointless waste of CPU time.

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