1- For n belongs to N (set of natural numbers) let B(n) denote the number of digits used in the binary representation of n. For example
B(1) = 1;
B(2) = 2;
B(3) = 2;
B(4) = 3:
Find a closed formula for B(n) for an arbitrary n belongs to N.
2: Prove that if gcd(a, b) = d then a/d and b/d are relatively prime.
3- Find the smallest prime factor of (p1)(p2)...(pk) +1, where p1, p2, ... pk are the kth smallest primes for all positive integers k not exceeding 50.
1. B(n) is the smallest integer which is greater than or equal to log(base 2) n.
In another word, we can find some integer k, such that 2^(k-1)<n<=2^k, then B(n)=k.
2. Proof: if a/d and b/d are not relatively prime, then they have a common factor r>1. Then
we have a/d=rx, ...
Binary Representations and Prime Factors are investigated.