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Search: id:A100873
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| A100873 |
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Pseudotwinprimes: primes p such that p+2 divides p^(p+2)+2 and p+2 is composite. |
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+0
2
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| 645, 1105, 2701, 2821, 4681, 6601, 10261, 12801, 14491, 16705, 18721, 19951, 25761, 29341, 30121, 31609, 33153, 39865, 41041, 42799, 49141, 52633, 55245, 62745, 68101, 72885, 83665, 85489, 90751, 104653, 107185, 129889, 129921
(list; graph; listen)
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OFFSET
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3,1
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COMMENT
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Conjecture 1: If p and p+2 are prime (twin primes), then p+2 divides p^(p+2)+2. Compared to the 1517 twin primes less than 130000, there were 33 pseudoprime occurrences. Conjecture 2: If for a randomly chosen prime p, p+2 divides p^(p+2)+2, then there is a greater than 98% chance that p and p+2 are twin primes. The sequence also contains several Carmichael numbers. In addition, If we relax the condition that p is prime or just odd, we get A001567 341,561,645,1105,1387,1729,1905,2047.. Sarrus numbers.
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FORMULA
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For primes p if p+2 divides p^(p+2) + 2 then p+2 is likley to be prime. If p+2 is composite, then p+2 is a pseudotwinprime.
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EXAMPLE
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For prime p = 643, 645 divides 643^(645)+ 2 and 645 is composite.
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PROGRAM
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(PARI) twtotwp2(n, n2, k) = { local(x, y, x2, c); c=0; forprime(x=n, n2, x2=x+2; y=x^x2+k; if(y%x2==0&!isprime(x2), c++; print1(x+2", "); ); ); print(); print(c", "pitwin(n2)) } pitwins(n) = \The number of twin prime pairs <= n. { local(c, x); c=0; forprime(x=3, n, if(isprime(x+2), c++) ); return(c) }
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CROSSREFS
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Sequence in context: A135384 A061324 A089295 this_sequence A063844 A067845 A057942
Adjacent sequences: A100870 A100871 A100872 this_sequence A100874 A100875 A100876
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KEYWORD
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hard,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Jan 09 2005
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