Primes from concatenated perfects +/- 1

davar55 · 2017-07-24, 14:30

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.

Batalov · 2017-07-24, 14:45

Are these big enough for you?
http://stdkmd.com/nrr/6/66667.htm#prime_list

davar55 · 2017-07-24, 14:55

Yes that's cool. What I'm looking for is the numerically smallest such number
for a given number of digits.

axn · 2017-07-24, 16:27

13164036458569648337239753460458722910223472318386943117783728128\
13164036458569648337239753460458722910223472318386943117783728128\
13164036458569648337239753460458722910223472318386943117783728128\
64967

Edit: Bonus 199-digits
13164036458569648337239753460458722910223472318386943117783728128\
13164036458569648337239753460458722910223472318386943117783728128\
13164036458569648337239753460458722910223472318386943117783728128\
6629

EDIT2: 205-digits
13164036458569648337239753460458722910223472318386943117783728128\
13164036458569648337239753460458722910223472318386943117783728128\
13164036458569648337239753460458722910223472318386943117783728128\
2828288129

davar55 · 2017-07-25, 13:54

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?

axn · 2017-07-25, 16:15

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.

davar55 · 2017-07-26, 13:42

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?

axn · 2017-07-26, 13:53

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]

davar55 · 2017-07-27, 13:25

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...

axn · 2017-07-27, 14:28

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...

davar55 · 2017-07-27, 15:17

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...

2017-07-24, 14:30	#1
davar55 May 2004 New York City 2×29×73 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, 16:27	#4
axn Jun 2003 2²·3·421 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

2017-07-27, 14:28	#10
axn Jun 2003 2²·3·421 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...

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2017-07-24, 14:45	#2
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3⁶×13 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 108A₁₆ Posts	Yes that's cool. What I'm looking for is the numerically smallest such number for a given number of digits.

2017-07-25, 13:54	#5
davar55 May 2004 New York City 1000010001010₂ 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?

2017-07-26, 13:42	#7
davar55 May 2004 New York City 1000010001010₂ 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?

2017-07-27, 13:25	#9
davar55 May 2004 New York City 10212₈ 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, 15:17	#11
davar55 May 2004 New York City 1000010001010₂ 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...