User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/existence

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/integrand

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/evaluation

User:GFauxPas/Sandbox/Zeta2/FourierSeries/

User:GFauxPas/Sandbox/Zeta2/Informal Proof

Problem Set
All hints are welcome.


 * $(1):$ Prove Greatest Power of Two not Divisor. Use this fact to prove Harmonic Numbers not Integers.

Suggestion: Derive a contradiction that $n + \dfrac 1 2 = m$ for $n, m \in \Z$.

Knuth's Hint:

There is precisely one term whose denominator is $2^k$, so $2^{k-1} H_n - \dfrac 1 2$ is a sum of terms involving only odd primes in the denominator. If $H_n$ were an integer, $2^{k-1} H_n - \dfrac 1 2$ would have a denominator equal to $2$."


 * $(2):$ Prove that:


 * $\displaystyle K_n = 1 + \frac 1 3 + \frac 1 5 + \ldots + \frac 1 {2n+1}$

is not an integer for $n > 1$.

Hint: The proof will be similar to the solution to $(1)$:

Consider the set $\omega = \left \{ {1, 3, \ldots, 2n+1}\right\}$ and let $3^d$ be the highest power of $3$ in $\omega$. Prove that $3^d$ is not a divisor of any other integer in $\omega$ and use this fact to prove $K_n$ is not an integer.


 * $(3):$ Prove that $\dfrac {\ln 2}{\ln 3}$ is irrational. For integers $p$ and $q$, what condition is essential for $\dfrac {\ln p}{\ln q}$ to be irrational? Justify your claims.


 * $(4):$ Use Maple to execute the command below and carefully explain the output.




 * $(5):$ Explain the behavior of the Euler Phi Function in finding all positive solutions to:


 * $(a): \phi(n) = 6, (b): \phi(n) = 14, (c):\phi(n) = 24$.