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?