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Search: id:A087525
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| A087525 |
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Primes p with the property that p-q does not divide p+q for all primes q < p. |
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1
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| 5, 7, 7, 11, 11, 11, 11, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Theorem: If z>y are primes then z-y divides z+y iff z=y+2. Proof: Let (1) z+y = (z-y)k for some integer k (2) z=y+2m. We must add multiples of 2 to y in order to avoid z=even. Substituting (2) into (1) we get y+2m + y = (y+2m-y)k 2y+2m = 2mk y+m = mk y/m + 1 = k for k to be an integer m must be 1 or y. If m = y then k=2 and z+y = 2*(z-y) z+y = 2z-2y 3y = z contradicting z > y. Therefore m=1 and z = y+2m = y+2 as desired. If z-y does divide z+y we have z the sequence of upper bound twin primes.
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PROGRAM
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(PARI) zmy(n) = { forprime(z=1, n, forprime(y=1, z-1, v1=z-y; v2=z+y; if(v2%v1<>0, print1(z", ")) \ if(v2%v1==0, print1(z", ")) yields sequence of twin primes ) ) }
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CROSSREFS
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Sequence in context: A086056 A137755 A053672 this_sequence A078496 A143699 A114521
Adjacent sequences: A087522 A087523 A087524 this_sequence A087526 A087527 A087528
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KEYWORD
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easy,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Oct 23 2003
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EXTENSIONS
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Corrected by Franklin T. Adams-Watters and T. D. Noe (noe(AT)sspectra.com), Oct 25 2006
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