special quadratic polynomials : f(n)=an²+bn+1

[bhelmes]

A peaceful night for you,


I noticed for some quadratic polynomial, such as f(n)=n²+1, f(n)=2n²-1, f(n)=2n²+1 and f(n)=4n²+1 you can make a linear substitution with n=p*k+n0 with p|f(n) and p=f(n0) and a division by p so that you get
f(k)=ak²+bk+1


This seems to be something special. Do they have a mathematical name and which mathematician has investigated them ?


A short link would be really nice from you,


Greetings from Corona-times,

I live only in the night like a vampir

[bhelmes]

A peaceful and pleasant night,

what is the difference between eliptic curves and quadratic polynomials,
resp. what mathematical property does eliptic curves have in opposite to quadratic polynomials, especially for the factorisation.

In my opinion both are double periodically function, for both they are suitable with complex numbers and the "Ordnung" (order) is "flexible".

Would be nice if someone gives me a mathematical hint.

Greetings from the factorisation side
Bernhard

@Batalov: do you choose your avatar gif for every mail or is it random

[Nick]

See "Factoring integers with elliptic curves" by H.W. Lenstra