Prime numbers and the Riemann hypothesis · more multiplicatively even numbers less than or equal...
Transcript of Prime numbers and the Riemann hypothesis · more multiplicatively even numbers less than or equal...
PRIME NUMBERS AND THE RIEMANN HYPOTHESIS
Part 1: What Are Primes and Other Questions
WHAT ARE THE PRIME NUMBERS?
Historically
■ Studied by philosophers and mathematicians for many centuries
– Pythagoras (precursor to the study of primes, perfect/amicable)
– Euclid (infiniteness of primes)
– Philolaus
– Aristotle
– Eratosthenes (sieves)
Specifically
■ Atoms
– “Indecomposable”
■ Defined
– Whole number greater than one that cannot be factored into a product of 2 smaller whole numbers
■ Factorization to Primes
– All numbers > 1
– Unique
Big Primes &Scope of the Set
■ Proof of infiniteness of primes (from our
book)
– Coincidentally also Euclid’s method
Book’s Largest VS
Largest Now
Still infinitely more…
Named Primes (Two Kinds)
Mersenne PrimesType: 2𝑛-1
Fermat PrimesType: 2𝑛 +1
The Sieve of Eratosthenes
The Sieve Up to 100
The Sieve Up to 100
The Sieve Up to 100
The Sieve Up to 100
Questions About Primes
■ Book presents many questions
– Yitang Zhang’s Proof
– Proof of infinitely many consecutive primes differing by no more than 70,000,000
■ Focus on Gaps
– In a continuation of Zhang’s ideas.
– Only consider gaps equal to even numbers
Popularity of gaps that are multiples of 6
QUESTION: Notice that gaps of size 6, 12,
18, etc. Why are multiples of 6 so
popular? See source (pg.2).
• All primes p>3 are congruent to 1 or 5.
• Dirichlet result on equidistribution of
primes of certain type 𝑝 ≡ 𝑎 (𝑚𝑜𝑑 𝑞).
Racing Gaps■ Which gaps grow faster as our X
tends toward infinity
– Not known as of yet
Racing Multiplicatively Even and Odd Numbers
■ Definition
– Even
– Odd
■ Counting multiplicatively even and
odd numbers ≤ 16
“Is there some X ≥ 𝟐 for which there are more multiplicatively even numbers less than or equal to X than multiplicatively odd ones?”■ We know that there indeed is some
X for which this is true.
– Lehman and Tanaka
■ The book mentions that if this were false, it would imply the Riemann Hypothesis
– Exercise: Why would this imply the Riemann Hypothesis?