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Math Problems

13 August
Posted in Article, Math Problems

Analytical Fibonacci

We derive Binet’s equation for the nth Fibonacci number as

     $$ F_n = \frac{1}{\sqrt 5}\left( \left(\frac{1+\sqrt 5}{2}\right)^n -  \left(\frac{1-\sqrt 5}{2}\right)^n  \right) $$

13 August
Posted in Article, Math Problems

Linear Recursive Sequence

We derive a general technique for solving full rank linear recursive sequences. Formally, a kth linear recursive sequence is defined as

     $$ x_n = a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3} + \stackrel{k}{\ldots} + a_k x_{n-k} $$

12 August
Posted in Article, Math Problems

Odd Coprimes

Let a be an odd integer. Prove that a^{2^n} + 2^{2^n} and a^{2^m}+2^{2^m} are relatively prime for all distinct positive integers n and m.

12 August
Posted in Article, Math Problems

More Prime Congruences

Find all primes p and q such that p+q = (p-q)^3

11 August
Posted in Article, Math Problems

Prime Congruence Class

Show that there are infinitely many primes of the form 4k - 1.

10 August
Posted in Article, Math Problems

Just the Evens

Find the sum of even positive divisors of 100000

10 August
Posted in Article, Math Problems

Sum of Divisors

Suppose we define the function \sigma: \mathbb{N} \to \mathbb{N} as the sum of the divisors of n

     $$ \sigma(n) = \sum_{d|n} d $$

Express \sigma(n) in terms of n‘s prime factorization

9 August
Posted in Article, Math Problems

That Other Little Gauss Story

Determine the product of distinct positive integer divisors of n = 420^4.

9 August
Posted in Article, Math Problems

Congruence by L.C.M

Determine the number of ordered pairs of positive integers (a,b) such that their least common multiple [a,b] = 2^35^711^{13}

9 August
Posted in Article, Math Problems

Randomly Chosen Divisors

Compute the probability that a randomly chosen positive divisor of 10^{99} is an integer multiple of 10^{88}