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A100875
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Pseudoquadprimes: p+4 for primes p where p+4 divides p^(p+4) + 4 and p+4 is composite.
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0
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15, 341, 435, 561, 645, 1905, 8321, 9131, 9605, 14351, 18705, 33153, 33227
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,1
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COMMENTS
| 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.
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FORMULA
| 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.
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EXAMPLE
| 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
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PROG
| (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", ") }
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CROSSREFS
| Sequence in context: A053104 A114937 A157965 * A034975 A012787 A030049
Adjacent sequences: A100872 A100873 A100874 * A100876 A100877 A100878
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KEYWORD
| nonn
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AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com), Jan 09 2005
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