Combinatorics | Number Theory Answers

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

Find all positive integers n such that n^4 - 1 is divisible by 5

Find all positive integers n such that n^4-1 is divisible by 5.

Find all positive integers n such that n^4 -1
is divisible by 5.

Calculate GCD(8,35) using EA (Euclid's Algorithm) and calculate the multiplicative inverse of 8 E Z3,, if it exists Also try the same for 21 E Z35.

Prove that there are infinitely many prime numbers of the form 6n + 5. Hint: any prime number p > 3 has the form p = 6n + 1 or p = 6n + 5 for some integer n. Use this fact and the fact that the product of two numbers of the form 6n + 1 has the same form.

Tom, Michael and Jane visit the same sport club. Tom visits the club every 5th day, Michael every 6th day and Jane every 8th day. If all three of them were in the sport club on August 5th, Sunday, what will be the next day when all of them will be there?

Let a = 3333333333^7777777777 (yes, there are ten digits ”3” and ten digits ”7”). If a is
written in decimal notation, what are the last two digits?
Hint: You need to compute a mod 100. Use Euler’s theorem and the fact that φ(100) = 40.

Let p and q be distinct prime numbers. Prove that p
q−1 + q
p−1 ≡ 1 mod pq.

Let p be an odd prime number, so that p = 2k + 1 for some positive integer k. Prove that
(k!)2 ≡ (−1)k+1 mod p.
Hint: Try to see how to group the terms in the product
(p − 1)! = (2k)! = 1 ∗ 2 ∗ 3 · · ·(2k − 2) ∗ (2k − 1) ∗ (2k)
to get two products, each equal to k! modulo p.