Combinatorics | Number Theory Answers

Let f(x) be a function on the set of real numbers x ≥ 1.
Define the function g(x) for x ≥ 1 by
g(x) = X
1≤n≤x
f(x/n).

Prove that
f(x) = X
1≤n≤x
µ(n)g(x/n).

Here µ is the M¨obius function.

Recall that ν(n) is the divisor function: it gives the number of positive divisors of n. After knowing this, prove that ν(n) is a prime number if and only if n = p^q−1, where p and q are prime numbers.

If f is a multiplicative arithmetic function then prove that
f([m, n])f((m, n)) = f(m)f(n)
for all positive integers m and n.
Here [m, n] is the least common multiple of m and n and (m, n) is the
greatest common divisor of m and n.

In how many ways can ball bearings labeled A, B, C and D be
shared by Jacob and Helen in such a way that
1 Jacob gets one ball bearing and Helen gets three
2 They each get two ball bearings?

(a) In how many different ways can the letters of the word wombat be arranged?
(b) In how many different ways can the letters of the word wombat be arranged if the letters wo
must remain together (in this order)?
(c) How many different 3-letter words can be formed from the letters of the word wombat? And
what if w must be the first letter of any such 3-letter word?

The 10 students who completed a special flying course are waiting to see if they will be awarded the one distinctiin or the one merit award for their offers.
In how many ways can the two awards be given if the same student can recieve both awards?
Show full working and the table for the sample space!

mae has more than 3 bracelets she has an even number of bracelets. Is the nimber of braceletstha mae has a prime number or compsite number ?

The sum of divisor function σ(n) returns the sum of all divisors d of the number n:
σ(n) = X
d|n
d
We denote Nk any number that fulfils the following condition:
σ(Nk) ≥ k · Nk
Find examples for N3, N4, N5 and prove that they fulfil this condition.

1.If m={prime number from 5—30}. Write out all the subjects of m

2. Given that k={even number from 1–10}. Find n(k)

1) in your own words explain collatz conjecture. Have this conjecture been proven?
2) What is the C(n) cycle and the T(n) cycle of the number n= 48?
3) Find the binary encoding of n= 32,53, 80 and explain why they all start with ''111''.
4) What is more common according to the data: r-curves with finite girth or acyclic r-curves?
REFERENCE ARTICLE link: https://arxiv.org/pdf/1811.00384.pdf