20220207, 03:34  #34 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2×1,153 Posts 
Hats off to you Dr. S.
Not quite sure how you got the conclusion, but while there are plenty of "TwinTwinTwinTwin" patterns based on a distance of 210 which are not divisible by any prime less than 47 (likely much higher) including 19, there is no such pattern that will not have at least one element divisible by 11. Dirty but sufficient code. Code:
\\EJD100A theFactorial = 47! \\\Removing any of these 11's will fail to yield results forprime(p=7503957281,19^1900,{ if(gcd(p+2,theFactorial )<2, if(gcd(p+6,theFactorial )<2 && gcd(p+8,theFactorial )<2, if(gcd(p+30,theFactorial )<2 && gcd(p+32,theFactorial )<2 && gcd(p+36,theFactorial )<2 && gcd(p+38,theFactorial )<2, if (gcd(p+210,theFactorial )<2 && gcd(p+212,theFactorial )<2 /*&& gcd(p+216) && gcd(p+218) && gcd(p+240) && gcd(p+242) && gcd(p+246) && gcd(p+248)*/, if( gcd(p+216,theFactorial)<2 && gcd(p+218,theFactorial)<2 && gcd(p+240,theFactorial)<2 && gcd(p+242,theFactorial)<2 && gcd(p+246,theFactorial)<2 && gcd(p+248,theFactorial)<2, print("TwinTwinTwinTwin"); print(p); ); ); ); ); ); }) ETA: 420 on the other hand would work: Code:
\\EJD110A theFactorial = 47! \\Removing the 11's will work for a distance of 420 forprime(p=7503957281,19^1900,{ if(gcd(p+2,theFactorial )<2, if(gcd(p+6,theFactorial )<2 && gcd(p+8,theFactorial )<2, if(gcd(p+30,theFactorial )<2 && gcd(p+32,theFactorial )<2 && gcd(p+36,theFactorial )<2 && gcd(p+38,theFactorial )<2, if (gcd(p+420,theFactorial )<2 && gcd(p+422,theFactorial )<2 , if( gcd(p+426,theFactorial)<2 && gcd(p+428,theFactorial)<2 && gcd(p+450,theFactorial)<2 && gcd(p+452,theFactorial)<2 && gcd(p+456,theFactorial)<2 && gcd(p+458,theFactorial)<2, print("TwinTwinTwinTwin"); print(p); ); ); ); ); ); }) Last fiddled with by a1call on 20220207 at 03:53 
20220207, 13:36  #35  
Feb 2017
Nowhere
6,101 Posts 
Quote:
Code:
? v=[0,2,6,8,30,32,36,38];w=vector(#v,i,x+ v[i]);w2=vector(#v, i,x + v[i]+210);f=prod(i=1,#v,w[i]*w2[i]);forprime(p=2,16,fp=Mod(1,p)*f;if(Mod(fp,x^px)==0,print(p))) 11 ? v=[0,2,6,8,30,32,36,38];w=vector(#v,i,x+ v[i]);w2=vector(#v, i,x + v[i]+420);f=prod(i=1,#v,w[i]*w2[i]);forprime(p=2,16,fp=Mod(1,p)*f;if(Mod(fp,x^px)==0,print(p))) ? 

20220211, 08:43  #36 
Jun 2003
Suva, Fiji
2^{3}×3×5×17 Posts 

20220211, 11:16  #37 
Jun 2003
Suva, Fiji
2^{3}×3×5×17 Posts 
I thought I would look at differences between 4twin constellations, (4 twins in 33 integers) using the first pattern listed in mart_r's list, post #22 on this thread. The smallest I have found to date (checked to 1.43e15) is:
9900 between (start) 736931653722599 and (start) 736931653712699 I think differences need to be 0 mod 30 The largest difference I found to date 80503603290 between 611475747027779 and 611395243424489 I'm also looking at patterns 2 and 3. Stop Press: Impressive closeness for pattern 2 shown by 2310 between 3577041656777 3577041654467 Last fiddled with by robert44444uk on 20220211 at 11:35 
20220211, 12:33  #38 
Jun 2003
Suva, Fiji
7F8_{16} Posts 
Wow, only 210 separate these two  pattern 2, (4 twins in 33)
200595358412147 200595358411937 I wonder if this is the closest two can get? Also a slightly large gap (87529363350) from the pattern 1's 1680433825465910 1680346296102560 Last fiddled with by robert44444uk on 20220211 at 12:34 
20220211, 13:16  #39  
May 2018
2^{2}·3·23 Posts 
Quote:
By the way, the smallest admissible distance between 2 consecutive sets of twin twin twin twin twin primes is 2118270. Therefore, we have the sequence 2, 6, 30, 420, 2310, 2118270, ... I wonder what the next term is. 

20220211, 14:03  #40  
Feb 2017
Nowhere
6,101 Posts 
Quote:
Quote:
Calling these patterns one, two, and three, I found that p + one and p + 192 + two together form an admissible 16tuple; that is, if the prime ktuples conjecture is true (and if my routine was writ right), there are infinitely many p for which all the following are prime. p+{0,2,12,14,24,26,30,32} and p+{192, 194, 204, 206, 210, 212, 222, 224} EDIT: My routine only looked at mixing and matching different patterns, and quit after its first "hit." I revised it to include "same same" pairs and to list all "hits." The line "1 2 192" is the previously mentioned result. 1 1 180 1 1 210 1 2 192 1 3 204 2 1 198 2 2 210 2 3 192 3 1 186 3 2 198 3 3 180 3 3 210 Last fiddled with by Dr Sardonicus on 20220211 at 14:30 

20220213, 16:47  #41 
Jun 2003
Suva, Fiji
11111111000_{2} Posts 
I actually found an overlapping set from the 3rd pattern!
Code:
1135141716537970+1 is 3PRP! (0.0000s+0.0002s) 1135141716537970+3 is 3PRP! (0.0000s+0.0002s) 1135141716537970+7 is 3PRP! (0.0000s+0.0001s) 1135141716537970+9 is 3PRP! (0.0000s+0.0002s) 1135141716537970+19 is 3PRP! (0.0000s+0.0001s) 1135141716537970+21 is 3PRP! (0.0000s+0.0002s) 1135141716537970+31 is 3PRP! (0.0000s+0.0001s) 1135141716537970+33 is 3PRP! (0.0000s+0.0002s) 1135141716537970+31 is 3PRP! (0.0000s+0.0001s) 1135141716537970+33 is 3PRP! (0.0000s+0.0002s) 1135141716537970+37 is 3PRP! (0.0000s+0.0002s) 1135141716537970+39 is 3PRP! (0.0000s+0.0003s) 1135141716537970+49 is 3PRP! (0.0000s+0.0002s) 1135141716537970+51 is 3PRP! (0.0000s+0.0002s) 1135141716537970+61 is 3PRP! (0.0000s+0.0002s) 1135141716537970+63 is 3PRP! (0.0000s+0.0002s) 
20220214, 14:43  #42 
Feb 2017
Nowhere
6,101 Posts 
As an exercise, I worked out the possibilities for p (mod 30030) for which p+{0,2,12,14,24,26,30,32} and p+{0,2,12,14,24,26,30,32}+ 180; and p+{0,2,6,8,18,20,30,32}, p+{0,2,6,8,18,20,30,32} + 180
are all relatively prime to 30030 = 13#. p+{0,2,12,14,24,26,30,32} and p+{0,2,12,14,24,26,30,32}+ 180 [p, p + 2, p + 12, p + 14, p + 24, p + 26, p + 30, p + 32, p + 180, p + 182, p + 192, p + 194, p + 204, p + 206, p + 210, p + 212] p == 827, 10067, 14687, or 16997 (mod 30030) p+{0,2,6,8,18,20,30,32}, p+{0,2,6,8,18,20,30,32} + 180 [p, p + 2, p + 6, p + 8, p + 18, p + 20, p + 30, p + 32, p + 180, p + 182, p + 186, p + 188, p + 198, p + 200, p + 210, p + 212] p == 12821, 15131, 19751, or 28991 (mod 30030) 
20220218, 18:10  #43 
Jun 2003
Suva, Fiji
2^{3}·3·5·17 Posts 
Small gaps between two sets of 6 twins of the same pattern do not look very likely, after a week of searching the best I could manage was
Between 1003698437366279 and 1005770184693929 the gap is "only" 2071747327650 Last fiddled with by robert44444uk on 20220218 at 18:11 
20220219, 14:00  #44 
May 2018
2^{2}×3×23 Posts 
Here are the patterns for the gaps between twin twin...twin primes. The sequence is 2, 6, 30, 420, 2310, 2118270, 338447078970, ...
Code:
2 [0, 2] 6 [0, 2, 6, 8] 30 [0, 2, 6, 8, 30, 32, 36, 38] 420 [0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458] 2310 [0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458, 2310, 2312, 2316, 2318, 2340, 2342, 2346, 2348, 2730, 2732, 2736, 2738, 2760, 2762, 2766, 2768] 2118270 [0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458, 2310, 2312, 2316, 2318, 2340, 2342, 2346, 2348, 2730, 2732, 2736, 2738, 2760, 2762, 2766, 2768, 2118270, 2118272, 2118276, 2118278, 2118300, 2118302, 2118306, 2118308, 2118690, 2118692, 2118696, 2118698, 2118720, 2118722, 2118726, 2118728, 2120580, 2120582, 2120586, 2120588, 2120610, 2120612, 2120616, 2120618, 2121000, 2121002, 2121006, 2121008, 2121030, 2121032, 2121036, 2121038] 338447078970 [0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458, 2310, 2312, 2316, 2318, 2340, 2342, 2346, 2348, 2730, 2732, 2736, 2738, 2760, 2762, 2766, 2768, 2118270, 2118272, 2118276, 2118278, 2118300, 2118302, 2118306, 2118308, 2118690, 2118692, 2118696, 2118698, 2118720, 2118722, 2118726, 2118728, 2120580, 2120582, 2120586, 2120588, 2120610, 2120612, 2120616, 2120618, 2121000, 2121002, 2121006, 2121008, 2121030, 2121032, 2121036, 2121038, 338447078970, 338447078972, 338447078976, 338447078978, 338447079000, 338447079002, 338447079006, 338447079008, 338447079390, 338447079392, 338447079396, 338447079398, 338447079420, 338447079422, 338447079426, 338447079428, 338447081280, 338447081282, 338447081286, 338447081288, 338447081310, 338447081312, 338447081316, 338447081318, 338447081700, 338447081702, 338447081706, 338447081708, 338447081730, 338447081732, 338447081736, 338447081738, 338449197240, 338449197242, 338449197246, 338449197248, 338449197270, 338449197272, 338449197276, 338449197278, 338449197660, 338449197662, 338449197666, 338449197668, 338449197690, 338449197692, 338449197696, 338449197698, 338449199550, 338449199552, 338449199556, 338449199558, 338449199580, 338449199582, 338449199586, 338449199588, 338449199970, 338449199972, 338449199976, 338449199978, 338449200000, 338449200002, 338449200006, 338449200008] 
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