### Primitive Friendly Pairs from a Formula

A formula generates some primitive friendly pairs = pfps .
This is a sequel to Primitive Friendly Integers and Exclusive Multiples where some terms are defined .

```If all of the following are prime :

p
prime_1  =  2 - 1

p+2
prime_2  =  2   - 1

p+2
2   + 1
prime_3  =  -------
3

then  p and p+2  are a pair of twin primes and

2p-1                   2p+3
2    * prime_1   and   2    * prime_2 * prime_3   form a pfp

p
2  + 1          1
with abundancy =  ------  =  2 + ----
p-1           p-1
2             2

In the formula, cases  p = 3 , 5 , and 17  yield :

pf1 pf1                          pf2 pf2             abund ( pf )
224 5001                      174592 90001000001     3*3    /    4
15872 90000000001             44736512 D       ;43;127 3*11   /   16
1125891316908032 X;131071 12592977287606574252032 b;174763;524287 3*43691/65536
```

prime_1 and prime_2 are Mersenne primes generated by a pair of twin primes .
How does the frequency of occurrence of these pfps compare to that of Fermat primes ?

```Footnote :
{ X;131071           ;  b;174763;524287   } multiplied by
{ 0A001;23;107;3851  ;  04003;31;61       }
produces a pfp found by comparing 2 4-friendlies ,
i.e. , 2 multiperfects with abundancy = 4 ,
and removing common factors which occur with like multiplicities .
2^33 3^10 7 11 23 83 107 331 3851 43691 131071
2^37 3^4 7 11^3 31 61 83 331 43691 174763 524287
XA0110001;83;107;331;3851;43691;131071
b4013;31;61;83;331;43691;174763;524287
XA0010001;107;3851;131071  b4003;31;61;174763;524287
```

2000 Mathematics Subject Classification : Primary 11A25 ; Secondary 11A05 , 11A41 , 11A51 , 11N25

Walter Nissen
2008-07-13