"Substract one and add one" algorithm

*To*: mathgroup at smc.vnet.net*Subject*: [mg59041] "Substract one and add one" algorithm*From*: "Gilmar" <gilmar.rodriguez at nwfwmd.state.fl.us>*Date*: Wed, 27 Jul 2005 01:25:40 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Dear Mathematica Users Forum Friends: I want to build a function h[n] that does the following: For n even greater and equal than 4: Case 1: If m=n/2 is prime then h[n]={n/2,n/2}. Done. Case 2: If m =n/2 is not prime; let p[1]=n/2 -1 and q[1]=n/2+1. If both p[1], and q[1] are prime then, h[n]={p[1],q[1]}. Done. If either one or both p[1] and q[1] are not prime; let p[2] =p[1]-1, and q[2]=q[1]+1. If both p[2], and q[2] are prime then h[n]={p[2],q[2]}. Done. If either one or both p[2] and q[3] are not prime; let p[3] =p[2]-1, and q[3]=q[2]+1. etc. I want to test empirically that a value h[n] = {p[k],q[k]} (for an appropriate integer k) exists. A few examples: n=4 n/2=2 is prime; so h[4]={2,2}. n=6 n/2=3 is prime; so h[6]={3,3}. n=8 is not prime; so p[1]=n/2 -1 =3 is prime and q[1]=n/2+1=5 is prime; so h[8]={3,5}. n=10 n/2=5 is prime; so h[10]={5,5}. n=12 n/2=6 is not prime; so p[1]=n/2-1=5 is prime and q[1]=n/2+1=7 is prime; so h[12]={5,7}. n=14 n/2=7 is prime; so h[14]={7,7}. n=16 n/2=8 is not prime; so p[1]=n/2-1=7 is prime but, q[1]=n/2+1=9 is not prime, so p[2]=7-1=6 is not prime, and q[2]=9+1=10 is not prime, so p[3]=6-1=5 is prime, and q[3]=10+1=11 is prime, so h[16]={5,11}. Thank you for your help!