a: \(A=2019\cdot2021\)
\(=\left(2020-1\right)\left(2020+1\right)\)
\(=2020^2-1\)
=B-1
=>A<B
b; \(B=\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\)
\(=\left(2-1\right)\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\)
\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\)
\(=\left(2^4-1\right)\left(2^4+1\right)\left(2^8+1\right)\)
\(=\left(2^8-1\right)\left(2^8+1\right)\)
\(=2^{16}-1\)
=A-1
=>B<A
c: \(A=4\left(3^2+1\right)\left(3^4+1\right)\cdot\ldots\cdot\left(3^{64}+1\right)\)
\(=\frac12\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)\cdot\ldots\cdot\left(3^{64}+1\right)\)
\(=\frac12\left(3^4-1\right)\cdot\left(3^4+1\right)\cdot\left(3^8+1\right)\cdot\ldots\cdot\left(3^{64}+1\right)\)
\(=\frac12\left(3^8-1\right)\cdot\left(3^8+1\right)\cdot\left(3^{16}+1\right)\cdot\left(3^{32}+1\right)\cdot\left(3^{64}+1\right)\)
\(=\frac12\left(3^{16}-1\right)\cdot\left(3^{16}+1\right)\cdot\left(3^{32}+1\right)\cdot\left(3^{64}+1\right)\)
\(=\frac12\left(3^{32}-1\right)\cdot\left(3^{32}+1\right)\cdot\left(3^{64}+1\right)\)
\(=\frac12\left(3^{64}-1\right)\cdot\left(3^{64}+1\right)=\frac12\left(3^{128}-1\right)=\frac12\cdot B\)
=>A<B
