Twin Prime Constellations

[a1call]

Hats off to you Dr. S.

Not quite sure how you got the conclusion, but while there are plenty of "Twin-Twin-Twin-Twin" 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.

\\EJD-100-A
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("Twin-Twin-Twin-Twin");
                        print(p);
                    );
                );
            );
        );
    );
})

So 210 as a "distance" won't do.


ETA: 420 on the other hand would work:

\\EJD-110-A
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("Twin-Twin-Twin-Twin");
                        print(p);
                    );
                );
            );
        );
    );
})

[Dr Sardonicus]

Originally Posted by a1call

<snip>
Not quite sure how you got the conclusion, but while there are plenty of "Twin-Twin-Twin-Twin" 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.
<snip>

The way I got the conclusion was to check whether the expressions x, x+2, etc always contain a complete residue system (mod p) for some prime p. Obviously, this can only be true if p is less than or equal to the number of expressions.

1. Multiply all the linear expressions x, x + 2 etc to get a polynomial f.
2. For each prime p <= the degree of the polynomial, take the reduction mod p, fp = Mod(1,p)*f
3. Check whether fp is divisible by x^p - x.
4. If it is, then f is divisible by p for every integer value of x; i.e. at least one of the linear expressions is always divisible by p.

In your latest example, this would work as follows:

? 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^p-x)==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^p-x)==0,print(p)))
?

[robert44444uk]

Originally Posted by robert44444uk

I'm up to 6e19 in the search for 9 twins (using all 4 patterns) in 105 digits without a result.

I took this up to at least 8.3e19 on all 4 possible sets, but the 2^64 barrier considerably slowed this down, so unlikely to take further.

[robert44444uk]

I thought I would look at differences between 4-twin 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

[robert44444uk]

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

[Bobby Jacobs]

Originally Posted by mart_r

You're right, I wasn't paying appropriate attention to the word "consecutive" in Bobby's post.

The currently largest known 16-tuplet has 35 digits, log is 78.56. The gap between the two double quadruplets is 382, which equals a merit of 4.86 in that region. (Also don't forget the possibility of up to two primes between the quadruplets.) Finding such a pattern without any prime in-between will be very hard, at least for now. At the least, it might be interesting to figure out the trade-off between number size and theoretical number of 16-tuplets to be found until the gap appears, for a possible future computation on a quantum chip.

It is OK to have primes in between. The two sets of twin prime quadruplets are consecutive sets of twin prime quadruplets. That is what I meant by "consecutive". However, there can be primes in between the sets of twin prime quadruplets, and even primes between the quadruplets. It is just a coincidence that the first set of twin prime quadruplets has no primes between the quadruplets.

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.

[Dr Sardonicus]

Originally Posted by robert44444uk

Wow, only 210 separate these two - pattern 2, (4 twins in 33)

200595358412147 200595358411937

I wonder if this is the closest two can get?
<snip>

I did some "mix-and-match" of the three patterns of four twins:

Originally Posted by mart_r

4 Twins:
p+{0,2,12,14,24,26,30,32}
p+{0,2,12,14,18,20,30,32}
p+{0,2,6,8,18,20,30,32}

I took separations of 33 or greater to make sure there was no overlap.

Calling these patterns one, two, and three, I found that

p + one and p + 192 + two

together form an admissible 16-tuple; that is, if the prime k-tuples 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

[robert44444uk]

I actually found an overlapping set from the 3rd pattern!

1135141716537970+1 is 3-PRP! (0.0000s+0.0002s)
1135141716537970+3 is 3-PRP! (0.0000s+0.0002s)
1135141716537970+7 is 3-PRP! (0.0000s+0.0001s)
1135141716537970+9 is 3-PRP! (0.0000s+0.0002s)
1135141716537970+19 is 3-PRP! (0.0000s+0.0001s)
1135141716537970+21 is 3-PRP! (0.0000s+0.0002s)
1135141716537970+31 is 3-PRP! (0.0000s+0.0001s)
1135141716537970+33 is 3-PRP! (0.0000s+0.0002s)

1135141716537970+31 is 3-PRP! (0.0000s+0.0001s)
1135141716537970+33 is 3-PRP! (0.0000s+0.0002s)
1135141716537970+37 is 3-PRP! (0.0000s+0.0002s)
1135141716537970+39 is 3-PRP! (0.0000s+0.0003s)
1135141716537970+49 is 3-PRP! (0.0000s+0.0002s)
1135141716537970+51 is 3-PRP! (0.0000s+0.0002s)
1135141716537970+61 is 3-PRP! (0.0000s+0.0002s)
1135141716537970+63 is 3-PRP! (0.0000s+0.0002s)

[Dr Sardonicus]

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)

[robert44444uk]

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

[Bobby Jacobs]

Here are the patterns for the gaps between twin twin...twin primes. The sequence is 2, 6, 30, 420, 2310, 2118270, 338447078970, ...

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]