This is a beautiful theorem which states that the difference between a positive integer and a prime power of , , is divisible by . It’s history can be found in Wikipedia: Fermat posited the theorem (and as usual, did not give the proof) while Euler first published a proof using mathematical induction. Below, we will state the theorem and provide a simple-to-understand proof using only modular arithmetic, followed by another simple proof using mathematical induction.
Continue reading “Fermat’s Little Theorem”
Riemann zeta function is a rather simple-looking function. For any number , the zeta function is the sum of the reciprocals of all natural numbers raised to the power.
Although the variable is a complex number, for the sake of simplicity, we will treat as real. (Real numbers are a subset of complex numbers.)
Continue reading “Euler’s Product Form of Riemann Zeta Function”