3,1

The 13 pseudoquadprimes listed is for primes less than 50000. There are 693 quadprimes less than 50000. So the chance is very good for prime p and p+4 to be quadprimes if p+4 divides p^(p+4) + 4. In general, if p and p+k are both prime then p+k divides p^(p+k)+k. If we do not know if p+k is prime and p+k divides p^(p+k) + k, then it is probable that p+k is prime. However, we get surprizes such as for k=64 we get 32 pseudo64primes less than 10000 while k=40 produces 4.

If p is prime and p+4 is prime then p and p+4 form a quad prime pair. In general, if p is prime and p+k is prime then p and p+k form a k difference prime pair. If p is prime and p+k divides p^(p+k) + k then it is likely that p+k is prime. If p+k is composite and divides p^(p+k) + k, then p+k is a pseudokprime.

p=7, p+4 = 11. (7^11+4)/11 = 179756977 so 11 prime, is not in the sequence

p=11,p+4 = 15. (11^15+4)/11 =278483211294377 so 15 composite is in the sequence

(PARI) ktokpk(n, n2, k) = { local(x, y, x2, c); c=0; forprime(x=n, n2, x2=x+k; y=x^x2+k; if(y%x2==0&!isprime(x2), c++; print1(x2", "); ); ); print(); print(c", ") }

Sequence in context: A053104 A114937 A157965 this_sequence A034975 A012787 A030049

Adjacent sequences: A100872 A100873 A100874 this_sequence A100876 A100877 A100878

nonn

Cino Hilliard (hillcino368(AT)gmail.com), Jan 09 2005