Arithmetic Proofs
Squared Numbers Equality Proof
Prove $\forall n,a\in\natnums$, $$ n^2 = (n+a)(n-a) + a^2 $$
Proof:
We can simply show the equalty holds true given some manipulation: $$ (n+a)(n-a) + a^2 \\ = n^2 -an +an -a^2 + a^2 \\ = n^2 \\ \square $$
Geometric Series Proof
Consider a series represented as such: $$ \Sigma_{n=0}^{n} ar^n = a + ar + ar^2 + ar^3 + … $$ The formula is: $$ S_n = a_1(\frac{1 - r^n}{1 - r}) $$
Proof: $$
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Karatsuba’s Algorithm (Multiplication)
Context
Kolmogorov’s Conjecture: Any algorithm for multiplying two $N$-digit numbers requires of the order of $N^2$ steps.