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Old 2017-07-24, 14:30   #1
davar55
 
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Default 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.
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Old 2017-07-24, 14:45   #2
Batalov
 
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Are these big enough for you?
http://stdkmd.com/nrr/6/66667.htm#prime_list
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Old 2017-07-24, 14:55   #3
davar55
 
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Yes that's cool. What I'm looking for is the numerically smallest such number
for a given number of digits.
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Old 2017-07-24, 16:27   #4
axn
 
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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
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Old 2017-07-25, 13:54   #5
davar55
 
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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?
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Old 2017-07-25, 16:15   #6
axn
 
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Quote:
Originally Posted by davar55 View Post
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.
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Old 2017-07-26, 13:42   #7
davar55
 
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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?
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Old 2017-07-26, 13:53   #8
axn
 
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Quote:
Originally Posted by davar55 View Post
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]
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Old 2017-07-27, 13:25   #9
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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...
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Old 2017-07-27, 14:28   #10
axn
 
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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...
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Old 2017-07-27, 15:17   #11
davar55
 
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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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