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.

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:

Code:

\\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);
);
);
);
);
);
})