Lecture Notes: The Beauty of Mathematics and Prime Numbers

Introduction

• Speaker: Tanya Cushman
• Topic: Mathematics, specifically prime numbers
• Personal background: • Former radio announcer from Sydney, Australia
• Anecdote about Australian culture and personal experiences at university

Early Love for Mathematics

• Began interest in mathematics in second grade at Boronia Park School
• Memorable incident with teacher Ms. Russell and the phrase "square peg through a round hole"
• Personal response sparked ridicule from classmates but solidified love for math

Mathematics as a Language of the Universe

• Math described as beautiful and natural, everywhere in the universe
• Quote by Descartes: "The universe is written in the mathematical language"

Introduction to Prime Numbers

• Explanation of prime numbers:
  • Not prime: 1 and any number that can be divided by other integers
  • Prime: Only divisible by 1 and itself (e.g., 7)
• Euclid's proof of infinite prime numbers
• Aim of the lecture: to explore the largest known prime number

Understanding Prime Numbers

• Basic explanation of exponents:
  • Example: 2 to the power of 5 = 2 x 2 x 2 x 2 x 2
• Examples of prime number forms:
  • 2^5 - 1 = 31 (prime)
  • 2^11 - 1 = 2047 (not prime, factors: 23 x 89)
  • 2^13, 2^17, 2^19 - all prime

Historical Context and Notable Mathematicians

• Leonhard Euler:
  • 18th-century mathematician, discovered 2^31 - 1 as the largest prime at the time
  • Proved primacy using traditional methods
• Dr. Curtis Cooper:
  • Current largest known prime: 2^57,885,161 - 17.5 million digits long
  • Uses networked computing to find large primes

Importance of Large Primes

• Large primes as a way to test computer speed and accuracy
• Connection to other scientific fields:
  • Similarities to RNA sequence unraveling and data searching (SETI)
• Collaboration between human minds and machines

Modern Discoveries and Highlights

• The excitement of discovering the Higgs boson
  • Peter Higgs's theoretical proposal
  • Confirmation of the theory within 50 years
• Reflection on the journey of human knowledge and exploration

Conclusion

• The essence of being human is to explore and understand the universe through math
• Quote: "We think, and therefore we are" - Descartes
• Final thoughts on the beauty and importance of mathematics and prime numbers

Personal Reflections

• Feelings about the intersection of mathematics with personal experiences and current events
• Encouragement to appreciate the beauty in numbers and mathematical concepts

Questions for Further Study

• Explore more about the properties of prime numbers
• Investigate the use of prime numbers in cryptography and computer science
• Understand the significance of the Higgs boson discovery in greater detail