mersenneforum.org Primes from concatenated perfects +/- 1
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 2017-07-24, 14:30 #1 davar55     May 2004 New York City 5·7·112 Posts Primes from concatenated perfects +/- 1 Since the first five perfect numbers are 6, 28, 496, 8128, and 33550336, the increasing sequence of numbers formed by concatenating perfect numbers begins 6 28 66 286 496 628 666 2828 2866 4966 6286 6496 6628 6666 8128 28286 ... Add or subtract one to generate an odd number hence possible prime. The particular problem is to find the smallest such 200-digit prime.
 2017-07-24, 14:45 #2 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2×4,931 Posts Are these big enough for you? http://stdkmd.com/nrr/6/66667.htm#prime_list
 2017-07-24, 14:55 #3 davar55     May 2004 New York City 5·7·112 Posts Yes that's cool. What I'm looking for is the numerically smallest such number for a given number of digits.
 2017-07-24, 16:27 #4 axn     Jun 2003 19·283 Posts 13164036458569648337239753460458722910223472318386943117783728128\ 13164036458569648337239753460458722910223472318386943117783728128\ 13164036458569648337239753460458722910223472318386943117783728128\ 64967 Edit: Bonus 199-digits 13164036458569648337239753460458722910223472318386943117783728128\ 13164036458569648337239753460458722910223472318386943117783728128\ 13164036458569648337239753460458722910223472318386943117783728128\ 6629 EDIT2: 205-digits 13164036458569648337239753460458722910223472318386943117783728128\ 13164036458569648337239753460458722910223472318386943117783728128\ 13164036458569648337239753460458722910223472318386943117783728128\ 2828288129 Last fiddled with by axn on 2017-07-24 at 16:36 Reason: Bonus2
 2017-07-25, 13:54 #5 davar55     May 2004 New York City 5×7×112 Posts Nice result for 200- 199- and 205- digits. Would it have been overly challenging if the puzzle had asked for all smallest solutions up to 250- or 300- digits?
2017-07-25, 16:15   #6
axn

Jun 2003

19·283 Posts

Quote:
 Originally Posted by davar55 Would it have been overly challenging if the puzzle had asked for all smallest solutions up to 250- or 300- digits?
A bit tedious, but not particularly challenging. The expected number of candidates to test for a 300 digit prime is about 350, which is not a lot. There are only 12 perfect numbers < 300 digits, so it is fairly straightforward to generate these candidates in lexicographical order.

 2017-07-26, 13:42 #7 davar55     May 2004 New York City 5×7×112 Posts Thanks. And once in lexicographic order the smallest prime might pop out immediately. Another issue: It's possible for there to be no prime satisfying the form for certain numbers of digits. Where is the first such gap,if any?
2017-07-26, 13:53   #8
axn

Jun 2003

124018 Posts

Quote:
 Originally Posted by davar55 Another issue: It's possible for there to be no prime satisfying the form for certain numbers of digits. Where is the first such gap,if any?
Code:
286 [7, 1; 41, 1] [3, 1; 5, 1; 19, 1]
496 [7, 1; 71, 1] [3, 2; 5, 1; 11, 1]
628 [17, 1; 37, 1] [3, 1; 11, 1; 19, 1]
666 [23, 1; 29, 1] [5, 1; 7, 1; 19, 1]

 2017-07-27, 13:25 #9 davar55     May 2004 New York City 5×7×112 Posts Nice and simple. I thought I had checked the 3-digits by hand ... oh well, that's the problem with checking by hand... As long as you have the program ... can you find the next such gap, if any? These gaps - the numbers of digits in them - might be few and far between, which might make the list somewhat interesting...
 2017-07-27, 14:28 #10 axn     Jun 2003 150116 Posts Here are the statistics for the first 20 digits: Code: 2 2 2 3 4 0 4 8 2 5 15 5 6 29 4 7 56 13 8 109 22 9 210 31 10 407 50 11 787 124 12 1524 199 13 2948 358 14 5705 617 15 11039 1144 16 21362 1922 17 41335 3454 18 79986 5984 19 154776 11350 20 299500 21390 digit, # of patterns, # of primes. As you can see, the number of primes just keep going up. So we don't expect there to be any further gaps. This is not a rigorous mathematical proof, of course, but...
 2017-07-27, 15:17 #11 davar55     May 2004 New York City 423510 Posts Really good, and convincing. The number of primes seems to approximately double for each digit, so maybe there's a proof possible in there somewhere...

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