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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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1
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15, 341, 435, 561, 645, 1905, 8321, 9131, 9605, 14351, 18705, 33153, 33227, 64821, 91001, 129921, 150851, 154101, 157641, 206601, 215265, 229503, 241001, 264773, 278693, 280601, 289941, 347721, 387731, 451905, 455295, 493697, 656601, 680627, 716141, 722261
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OFFSET
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1,1
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COMMENTS
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The 13 pseudoquadprimes listed are 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 surprises such as for k=64 we get 32 pseudo64primes less than 10000 while k=40 produces 4.
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LINKS
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FORMULA
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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
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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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MATHEMATICA
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lst = {}; Do[q = p + 4; If[! PrimeQ[q] && PowerMod[p, q, q] == p, AppendTo[lst, q]], {p, Prime@Range[2^16]}]; lst (* Arkadiusz Wesolowski, Jun 01 2013 *)
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PROG
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(PARI) ktokpk(n=1, n2, k=4) = { local(x, y, x2); forprime(x=n, n2, x2=x+k; y=x^x2+k; if(y%x2==0&!isprime(x2), print1(x2, ", "); ); ); }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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