[This is just the old draft of the page]
[I inadvertently removed the file]


Fermat noticed that when we restrict Mersenne numbers M(n)=2^n-1 to n=2^m
we reach to a [chain] of numbers, each number dividing its successor.
and then he extracted the [principal] of that chain by dividing each term by its previous one.
(2^2^(m+1)-1)/(2^2^m-1) = 2^2^m+1
which we call them Fermat Numbers F(m)=2^2^m+1.


chain      Phi1(2^2^m) = 2^2^m-1
principal  Phi2(2^2^m) = 2^2^m+1



chain      Phi1(2^p^n) , p odd prime
principal  Phi(p)(2^p^n)

Shanks noticed [when?] that
instead of powers of 2, we can restrict Mersenne numbers M(n) = 2^n-1 = Phi1(2^n)
to n=p^m, the powers of p, where p is an odd prime,
reaching to a [chain] of numbers Phi1(2^p^m) = 2^p^m-1
extracting their [principal] (2^p^(m+1)-1)/(2^p^m-1) = Phi(p^(m+1))(2) = Phi(p)(2^p^m)
proposed them as a further generalization for Fermat Numbers.

Phi3(2^3^m) =                                4^3^m +2^3^m +1
Phi5(2^5^m) =                 16^5^m +8^5^m +4^5^m +2^5^m +1
Phi7(2^7^m) = 64^7^m +32^7^m +16^7^m +8^7^m +4^7^m +2^7^m +1
...

Let's call them Shanks numbers



I have further generalized his idea 



chain      Phi2(2^p^m), p odd prime
principal  Phi(2p)(2^p^m)

 Phi6(2^3^m) =                                4^3^m -2^3^m +1
Phi10(2^5^m) =                 16^5^m -8^5^m +4^5^m -2^5^m +1
Phi14(2^7^m) = 64^7^m -32^7^m +16^7^m -8^7^m +4^7^m -2^7^m +1
...

Let's call them alternating Shanks numbers.




more alternating

chain      Phi(p)(2^2^n), p odd prime
principal  Phi(2p)(2^2^n)


 Phi6(2^2^m) =                                4^2^m -2^2^m +1   = Eight(m)?
Phi10(2^2^m) =                 16^2^m -8^2^m +4^2^m -2^2^m +1
Phi14(2^2^m) = 64^2^m -32^2^m +16^2^m -8^2^m +4^2^m -2^2^m +1
...



and the start of the real generalization

chain       Phi(q)(2^p^m), (p,q different odd primes) 
principal   Phi(pq)(2^p^m)

Phi15(a^3^m)  Phi21(a^3^m)  Phi35(a^5^m)
Phi15(a^5^m)  Phi21(a^7^m)  Phi35(a^7^m)

Phi15(2^3^m)  Phi21(2^3^m)  Phi35(2^5^m)
Phi15(2^5^m)  Phi21(2^7^m)  Phi35(2^7^m)



and their alternating form

chain       Phi(2q)(a^p^m), (p,q different odd primes) 
principal   Phi(2pq)(a^p^m)

Phi30(a^3^m)  Phi42(a^3^m)  Phi70(a^5^m)
Phi30(a^5^m)  Phi42(a^7^m)  Phi70(a^7^m)

Phi30(2^3^m)  Phi42(2^3^m)  Phi70(2^5^m)
Phi30(2^5^m)  Phi42(2^7^m)  Phi70(2^7^m)



chain        Phi(pq)(a^2^m)
principal    Phi(2pq)(a^2^m)

chain:     Phi15(a^2^m)  Phi21(a^2^m)  Phi35(a^2^m)  
principal: Phi30(a^2^m)  Phi42(a^2^m)  Phi70(a^2^m)  

Phi30(2^2^m)
Phi42(2^2^m)
Phi70(2^2^m)



[?]

Phi15(a^7^m)  Phi21(a^5^m)  Phi35(a^3^m)  
Phi30(a^7^m)  Phi42(a^5^m)  Phi70(a^3^m)