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MathGroup Archive 2005

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Re: "Substract one and add one" algorithm

  • To: mathgroup at smc.vnet.net
  • Subject: [mg59060] Re: "Substract one and add one" algorithm
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Thu, 28 Jul 2005 02:26:04 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, U.K.
  • References: <dc776m$k2h$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Gilmar wrote:
> 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!
> 
Hi Gilmar,

You could try something along the following lines:

In[1]:=
h[(n_)?(EvenQ[#1] && #1 >= 4 & )] := {n/2, n/2} /; PrimeQ[n/2]

In[2]:=
h[(n_)?(EvenQ[#1] && #1 >= 4 & )] := {n/2 - 1, n/2 + 1} /;
    PrimeQ[n/2 - 1] && PrimeQ[n/2 + 1]

In[3]:=
h[(n_)?(EvenQ[#1] && #1 >= 4 & )] := Module[{p, q},
    p = n/2 - 1; q = n/2 + 1; checkpq[p, q]]

In[4]:=
checkpq[(p_)?Positive, q_] := {p, q} /; PrimeQ[p] && PrimeQ[q]

In[5]:=
checkpq[(p_)?Positive, q_] := checkpq[p - 1, q + 1]

In[6]:=
h[2]

Out[6]=
h[2]

In[7]:=
h[4]

Out[7]=
{2, 2}

In[8]:=
h[5]

Out[8]=
h[5]

In[9]:=
h[6]

Out[9]=
{3, 3}

In[10]:=
h[8]

Out[10]=
{3, 5}

In[11]:=
h[10]

Out[11]=
{5, 5}

In[12]:=
h[12]

Out[12]=
{5, 7}

In[13]:=
h[14]

Out[13]=
{7, 7}

In[14]:=
h[16]

Out[14]=
{5, 11}

Best regards,
/J.M.


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