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Old 2022-02-07, 13:36   #35
Dr Sardonicus
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Feb 2017

2×11×277 Posts

Originally Posted by a1call View Post
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.
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)))
? 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)))
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