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Posts Tagged ‘ number theory ’

6 November

Annoying Mclaurin Series

Suppose that we’re given the function B(z) = \frac{1 + z}{1 - z - z^2}, find the ordinary generating function associated with it in the form of B(z) = \sum_{k=0}^\infty b_k k^z. Furthermore, find/compute \sigma_n = \sum_{k=0}^n b_k.

15 August
Posted in Article, Math Problems

Power Reduction in Congruences

Suppose you have integers a,b that are relatively prime to m such that

    $$ a^x \equiv b^x \mod m \hspace{4mm}\mbox{ and }\hspace{4mm} a^y \equiv b^y \mod m $$

then

    $$ a^{\gcd(x,y)} \equiv b^{\gcd(x,y)} \mod m $$

14 August
Posted in Article, Math Problems

Even Pascals

Let a=4k-1, where k is an integer. Prove that for any positive integer n the number

     $$ s_n = 1 - {n \choose 2}a + {n \choose 4}a^2 - {n \choose 6}a^3 + \cdots $$

is divisible by 2^{n-1}.

13 August
Posted in Article, Math Problems

Polynomial Divisors

Let p(x) = x^2 -3x + 2. Show that for any positive integers  n \ge 2 there exists unique numbers a_n, b_n such that the polynomial  x^n - a_n x - b_n is divisible by p(x)

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.