Gaporial primes

[enzocreti]

I wonder if somebody has ever thought about "gaporial" primes.
The record holders primes with maximum gap after it are: 2,3,7,23,89,113...

So why not to think at the gaporial primes?

2+1=3 is the first gaporial prime
2*3+1=7 is the second gaporial prime
2*3*7+1=43 is the third gaporial prime.
I have not a code for continuing so I stop here…

the next should be 2*3*7*23+1=967

[Dylan14]

Although it isn’t quite what you are considering, you may want to look at 081411 (partial product of all prime gaps) and consider finding a way to make primes from that sequence (not directly, of course, the only prime in that sequence is 2).

[Batalov]

Originally Posted by enzocreti

I wonder if somebody has ever thought about "gaporial" primes.
The record holders primes with maximum gap after it are: 2,3,7,23,89,113...

So why not to think at the gaporial primes?

2+1=3 is the first gaporial prime
2*3+1=7 is the second gaporial prime
2*3*7+1=43 is the third gaporial prime.
I have not a code for continuing so I stop here…

the next should be 2*3*7*23+1=967

and the next, 2*3*7*23*89+1 = 5^2 · 19 · 181, composite

[lukerichards]

Originally Posted by Batalov

and the next, 2*3*7*23*89+1 = 5^2 · 19 · 181, composite

2*3*7*23*89*113+1 = 9715063 = 103 * 94321

2*3*7*23*89*113+1 = 5080977427 is prime (checked on my Casio scientific calculator)

2*3*7*23*89*113*887+1 = 5*172345199 is composite

2*3*7*23*89*113*887*1129+1 = 17^2*313*401*268211 is composite

The next are:

11^2*17*6276243281279 Composite
229123*538163475229561 Composite
1163*1662770588367120449309 Composite
5*13*59*24091*13761561049*29824941343 Composite

2*3*7*23*89*113*523*887*1129*1327*9551*15683*19609*31397+1
= 1190571955160538884416660138998919 Prime

[lukerichards]

Having exhausted all entries of 002386, there are no other Gaporial primes with fewer than 154 digits.

[enzocreti]

Originally Posted by lukerichards

Having exhausted all entries of 002386, there are no other Gaporial primes with fewer than 154 digits.

Interesting to see that the last gaporial prime 1190571955160538884416660138998919*2-1 is prime!

Instead the first five gaporial primes have the property that p+4 is as well prime.

[enzocreti]

Originally Posted by lukerichards

2*3*7*23*89*113+1 = 9715063 = 103 * 94321

2*3*7*23*89*113+1 = 5080977427 is prime (checked on my Casio scientific calculator)

2*3*7*23*89*113*887+1 = 5*172345199 is composite

2*3*7*23*89*113*887*1129+1 = 17^2*313*401*268211 is composite

The next are:

11^2*17*6276243281279 Composite
229123*538163475229561 Composite
1163*1662770588367120449309 Composite
5*13*59*24091*13761561049*29824941343 Composite

2*3*7*23*89*113*523*887*1129*1327*9551*15683*19609*31397+1
= 1190571955160538884416660138998919 Prime

2*3*7*23*89*113*523*887*1129*1327*9551*15683*19609*31397+1 is NOT prime

[lukerichards]

Originally Posted by enzocreti

2*3*7*23*89*113*523*887*1129*1327*9551*15683*19609*31397+1 is NOT prime

My apologies! I calculated:
2 · 3 · 7 · 23 · 89 · 113 · 887 · 1129 · 1327 · 9551 · 15683 · 19609 · 31397 + 1

The rest of mine could be off too...

[Dr Sardonicus]

I note that a previous post seems to have missed the prime 523, with gap 18 before the next prime 541. The output of the following mindless Pari-GP script (with primelimit bumped up to 2^30) lists index, "gappy" prime of that index, the "champion" gap before the next prime, and, if the "gaporial" product + 1 is (pseudo)prime, an asterisk.

? m=0;oldp=2;gmax=0;pprod=1;forprime(p=3,1000000000,g=p-oldp;if(g>gmax,pprod*=oldp;gmax=g;m++;if(ispseudoprime(pprod+1),c="*",c="");print(m" "oldp" "g,c));oldp=p)

1 2 1*
2 3 2*
3 7 4*
4 23 6*
5 89 8
6 113 14
7 523 18*
8 887 20
9 1129 22
10 1327 34
11 9551 36
12 15683 44
13 19609 52
14 31397 72
15 155921 86
16 360653 96
17 370261 112
18 492113 114
19 1349533 118
20 1357201 132
21 2010733 148
22 4652353 154
23 17051707 180
24 20831323 210
25 47326693 220
26 122164747 222
27 189695659 234
28 191912783 248
29 387096133 250
30 436273009 282

[enzocreti]

1 2
2 3
3 7
4 23
5 89
6 113
7 523
8 887
9 1129
10 1327
11 9551
12 15683
13 19609
14 31397
15 155921
16 360653
17 370261
18 492113
19 1349533
20 1357201
21 2010733
22 4652353
23 17051707
24 20831323
25 47326693
26 122164747
27 189695659
28 191912783
29 387096133
30 436273009
31 1294268491
32 1453168141
33 2300942549
34 3842610773
35 4302407359
36 10726904659
37 20678048297
38 22367084959
39 25056082087
40 42652618343
41 127976334671
42 182226896239
43 241160624143
44 297501075799
45 303371455241
46 304599508537
47 416608695821
48 461690510011
49 614487453523
50 738832927927
51 1346294310749
52 1408695493609
53 1968188556461
54 2614941710599
55 7177162611713
56 13829048559701
57 19581334192423
58 42842283925351
59 90874329411493
60 171231342420521
61 218209405436543
62 1189459969825483
63 1686994940955803
64 1693182318746371
65 43841547845541059
66 55350776431903243
67 80873624627234849
68 203986478517455989
69 218034721194214273
70 305405826521087869
71 352521223451364323
72 401429925999153707
73 418032645936712127
74 804212830686677669
75 1425172824437699411
76 5733241593241196731
77 6787988999657777797

This is the vector of the prime gap records...who wants could find other gaporial primes with plus one ga#+1

[enzocreti]

It is interesting that the first five ga#+1 have the property that ga#+5 is prime as well