A087525 - OEIS

%I #11 Mar 15 2023 11:12:39

%S 5,7,7,11,11,11,11,13,13,13,13,17,17,17,17,17,17,19,19,19,19,19,19,23,

%T 23,23,23,23,23,23,23,29,29,29,29,29,29,29,29,29,31,31,31,31,31,31,31,

%U 31,31,37,37,37,37,37,37,37,37,37,37,37,41,41,41,41,41,41,41,41,41,41

%N Primes p with the property that p-q does not divide p+q for all primes q < p.

%C 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.

%o (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 ) ) }

%K easy,nonn

%O 1,1

%A _Cino Hilliard_, Oct 23 2003

%E Corrected by _Franklin T. Adams-Watters_ and _T. D. Noe_, Oct 25 2006