Prime Numbers and Shizits Nick Handrick, Chris Magyar University of Wisconsin Eau Claire March 7, 2017
Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers.
Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C.
Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE.
Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not.
Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not..
Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not.. Let EF not be prime; therefore it is measured by some prime number G.
Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not.. Let EF not be prime; therefore it is measured by some prime number G..
Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not.. Let EF not be prime; therefore it is measured by some prime number G.. Therefore, the prime numbers A, B, C, G have been found which are more than the assigned multitude of A, B, C.
Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes?
Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1)
Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2
Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2 Mark the smallest number (3), and remove all multiples of 3
Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2 Mark the smallest number (3), and remove all multiples of 3.
Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2 Mark the smallest number (3), and remove all multiples of 3. Once you reach N, every remaining number in your set is prime as well!
Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2 Mark the smallest number (3), and remove all multiples of 3. Once you reach N, every remaining number in your set is prime as well!
Prime Distribution / Probability range primes probability [1, 30] 10 1 3
Prime Distribution / Probability range primes probability 1 [1, 30] 10 3 1 [1, 100] 25 4
Prime Distribution / Probability range primes probability 1 [1, 30] 10 3 1 [1, 100] 25 4 [1, 1000] 168 1 6
Prime Distribution / Probability range primes probability 1 [1, 30] 10 3 1 [1, 100] 25 4 [1, 1000] 168 1 6 [1, 1000000] 78498 1 13
Prime Distribution / Probability range primes probability 1 [1, 30] 10 3 [1, 100] 25 1 4 [1, 1000] 168 1 6 [1, 1000000] 78498 1 13 [1, 10000000000] 455052512 1 22
Primes and Goldbach An interesting graph (not entirely related but still awesome) Number of ways even numbers can be expressed as the sum of two primes:
Primes in History Carl Friedrich Gauss (1824-1908)
Primes in History Carl Friedrich Gauss (1824-1908) G(x) X number of digits of X
x/ log x Yet another approximation for π(x) Gauss investigated π(x), and used some intuition to produce an approximation The intuition he had was that the odds some number X is prime is inversely proportional to the number of digits of X. π(x) X the number of digits of X X log X
Log Integral Function What is Li(x)? The Logarithmic Integral, also known as Li(x), is another mysterious way to approximate π(x) Definition: Li(x) = We ll see a graph of it soon x 2 1 log t dt
Here s that graph I promised
Here s that graph I promised The red staircase is π(x)
Here s that graph I promised The red staircase is π(x) The curve above it is Li(x)
Here s that graph I promised The red staircase is π(x) The curve above it is Li(x) The curve below it is x log x
Here s that graph I promised The red staircase is π(x) The curve above it is Li(x) The curve below it is x log x The graphs are a Good Fit approximation of each other
Good Fit approximation What does it even mean, rigorously?
Good Fit approximation What does it even mean, rigorously? They approach each other of course!
Good Fit approximation What does it even mean, rigorously? They approach each other of course! The log integral function converges much faster, but is more computationally intensive The Prime Number Theorem says that Li(x) and π(x) go to inifinity at the same rate
Riemann Hypothesis For any real number X, the number of prime numbers less than X is approximately Li(X).
Riemann Hypothesis For any real number X, the number of prime numbers less than X is approximately Li(X). Li(X) π(x) X log(x) for all X 2.01
Square Root Error Random error and random walks According to Barry Mazur, Statisticians use square root error as a gold standard for empirical data accuracy
ψ(x) and Riemann Hypothesis Second Formulation We talked earlier about ψ(x), but what does it have to do with the RH? ψ(x) = log p p k x = x ρ x ρ ρ log(2π) x where ρ are ALL zeroes of the zeta function You prove: ζ (s) ζ(s) = log(2π)