Shanks' generalization of Fermat numbers
(Shanks) noticed [when/where?] 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 part" (2^p^(m+1)-1)/(2^p^m-1) = Phi(p^(m+1))(2) = Phi(p)(2^p^m),
Where Phi(n)(x) is the n-th cyclotomic polynomial.

Phi2(2^2^m) = 2^2^m+1 = F(m) the original 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
...

I noticed that his idea is even working for composite p.
So we credit him with all the numbers of the form Phi(s)(2^s^m)
- where s stands for Shanks -
and call these numbers Shanks-Fermat numbers.
[really, no mathematician before him considered powers of p?!]

Phi6(2^6^m)   = Phi3(-2^6^m)  = 4^6^m-2^6^m+1
Phi10(2^10^m) = Phi5(-2^10^m) = 16^10^m-8^10^m+4^10^m-2^10^m+1
Phi12(2^12^m) = Phi6(4^12^m)  = Phi3(-4^12^m) = 16^12^m-4^12^m+1
Phi14(2^14^m) = Phi7(-2^14^m) = 64^14^m-32^14^m+16^14^m-8^14^m+4^14^m-2^14^m+1
Phi15(2^15^m) = 256^15^m-128^15^m+32^15^m-16^15^m+8^15^m-2^15^m+1
Phi18(2^18^m) = Phi6(8^18^m) = Phi3(-8^18^m) = 64^18^m-8^18^m+1
Phi20(2^20^m) = [...]
Phi21(2^21^m) = 4096^21^m-2048^21^m+512^21^m-256^21^m+64^21^m-16^21^m+8^21^m-2^21^m+1
...
Phi30(2^30^m)

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