Send the link below via email or IMCopy
Present to your audienceStart remote presentation
- Invited audience members will follow you as you navigate and present
- People invited to a presentation do not need a Prezi account
- This link expires 10 minutes after you close the presentation
- A maximum of 30 users can follow your presentation
- Learn more about this feature in our knowledge base article
Do you really want to delete this prezi?
Neither you, nor the coeditors you shared it with will be able to recover it again.
Make your likes visible on Facebook?
Connect your Facebook account to Prezi and let your likes appear on your timeline.
You can change this under Settings & Account at any time.
Transcript of Number-Theoretic Functions
is the set of positive integers is said to be a number-theoretic ( or arithmetic) function. Definition 1. Definition 2. If n=p_1^(k_1 ) p_2^(k_2 )…p_r^(k_r ) is the prime factorization of n>1,
then the positive divisors of n are
precisely those integers d of the form
d=p_1^(a_1 ) p_2^(a_2 )…p_r^(a_r ) Theorem 1. If n=12, find
τ(n) and σ(n). Example: Given a positive integer n, let τ(n) denote the number of positive divisors of n and σ(n) denote the sum of these divisors. ∑ f(d)
(d|n) Notation: ∑ f(d)
(d|20) Example: τ(n) = ∑ 1
σ(n)= ∑ f(d)
(d|n) Expressions n=1000 (written) Theorem 2. A number-theoretic function f is
said to be multiplicative if
whenever gcd(m,n)=1. Definition 3. σ(ab)= σ(a) σ(b)
τ(ab)=τ(a)τ(b) Multiplicative Property When n=957, show
that σ(n)=σ(n+1). Example: Verify that [τ(n)][τ(n+1)]=τ(n+3)
holds for n=3655. Try this! Thank