A formula for computing exponential sums

The expression $s(s^L)’$$ can be rewritten as: $$sum_{L=0}^M L^2s^L=0^2+1^2 s + s^2 left(sum_{L=2}^{M} s^L right)”+s left(sum_{L=2}^{M} s^L right)’$$ Another method is to manipulate the inner expression to resemble a derivative: $$ begin{align} sum_{L=0}^Mleft(Ls^{L-1}right)sL & =ssum_{L=0}^Mleft(partial_ss^Lright)L\ & =spartial_ssum_{L=0}^Ms^LL\ & =spartial_ssum_{L=0}^Mleft(Ls^{L-1}right)s\ & =spartial_sleft(ssum_{L=0}^Mleft(Ls^{L-1}right)right)\ & =spartial_sleft(ssum_{L=0}^Mpartial_ss^Lright)\ & =spartial_sleft(spartial_ssum_{L=0}^Ms^Lright)\ & =spartial_sleft(spartial_sfrac{s^{M+1}-1}{s-1}right)\ end{align}$$ After simplifying the above expression, apply the […]

What is the antonym of a prime number?

The number of checks required to verify the prime factors of $N-1$ is at most $O(log N)$, with each check being a modular exponentiation in $mathbb{Z}/Nmathbb{Z}$ taking $O(log^2 N)$ time. Therefore, the total complexity is $O(log^3 N)$. In $mathbb{Z}$, a prime is any non-zero, non-invertible element $p$ of the ring such that for all $a, […]

Total of the initial 100 natural numbers

Initially, breaking mathematical equality is required by claiming that a divergent series can be equal to a particular value. Although this is an enjoyable way to manipulate divergent series, it does not adhere to the rigorous principles of mathematical equality. Solution 1: Do you need to calculate the sum of $a$ plus all consecutive integers […]

Is -1(mod 13) equivalent to 12(mod 13)?

In $mathbb{Z}^*_{13}$, the group law is “multiplication modulo 13”, which includes all invertible elements modulo 13. The group law fulfills certain properties, such as being associative and having a neutral element and opposite element for each element in the group. The neutral element in this case is 0, and for example, 4 has an opposite […]