February 19

Plan:

  • Questions
  • Point to nice links in feedback from last class. Here are two more I found:
  • Change of plan: On to Chapter 4 (not 2). Chapter 2 content is nice, but there are too few historical passages to read.
  • Pythagorean triples
    • Babylon
    • Euclid (algebra without algebra)
    • Modern formulation first or second?
  • Fermat, Euler, Gauss, Wiles


Student Questions:


Discussion
The discussion page can be found here


Theorems/Ideas Names
Pythagorean Triples Ancient Babylonians
Perfect Numbers Diophantus
Mersenne Primes Bachet
Fermat's Theorems Euclid
Fundamental Thm. Arithmetic Fermat
Euler
Legendre
Gauss
Germain
Kummer
Wiles

Timeline (est.) (Mingzhi):

flickr:8490751189
  • Might we have a time line where the horizontal tick marks were accurately placed to scale? Ethan

Homework:

  • Discussed about the new assignment, the paper.
  • A number of questions for the new assignment are now available. The paper is the first question, but there are a number of other number theory questions. As of the date and time of this update, there is not an announced deadline.
  • Please click here to go to the assignment page.

Last Time:

  • Question: Are Pentagons in tiling regular? Able to put a poincare circle in it?

Moving Forward:

  • Talked about the reason we are doing Chapter 4 instead of 2.

Fundamental Concepts of Arithmetic/Some Interesting Discoveries:

  • Sum of no numbers is 0, product of no numbers is 1.
  • 91 looks like a prime, but it is not (13 times 7).

Fundamental Theorem of Arithmetic:
The Fundamental Theorem of Arithmetic:
Every number is a product of primes and can be factored uniquely


The following is a proof of the first half (every number is a product of primes)
Proof
Note that we are dealing with the set of natural numbers, so I will prove this using Peano's Fifth Axiom, The Principle of Mathematical Induction. More specifically, I will use strong induction

The Principle of Strong Mathematical Induction
Let S be a proper subset of the natural numbers, and let P(n) be a predicate statement. If $P_{a}, P_{b}, \dots, P[k]$ is true, and if for any natural number greater than k, if l is true for all $a\le l < n$ then n is also true, then we may conclude that P(n) is true for any natural number greater than a.

Note that we will use numbers that are at least 1. If n=1, n is a product of no primes and thus this is true. If n=2, n is a prime number and is thus a product of the prime 2.

Now, let n be an arbitrary number such that $n > 2$. Suppose that for all values less than n and greater than or equal to 2, those values are products of primes (either simply a prime or a product of primes). Then for n, we have two cases:

  • n is prime

If n is prime, then clearly n is the product of the prime n, making n a product of primes as required

  • n is not prime

If n is not prime, then it must be the case that there exists some values other than 1 and n that divide n by the properties of not being prime. Note that the values must be greater than 1 and less than n based on the previous statement, and considering that a value less than n would have to be multiplied by a value greater than 1 to produce n. We shall denote these values as a and b and note that $1 \le a < n$, $1 \le b < n$, and $ab=n$. Given that we know now that a and b are less than n and greater than or equal to 1, it must be the case that they fall into our inductive hypothesis, that is, that they are also products of primes (or are themselves primes). This, however, means that since a is a product of primes and b is a product of primes, and n is a product of a and b, n must be a product of the primes that make up a and b. Therefore, n is a product of primes.

As these two cases are exhaustive, covering all possible values of n in this set, we may conclude that n is a product of primes. Since n is arbitrary, this can be generalized to all $n \in S$.

Therefore, by strong induction, we may conclude that every number is a product of primes.
Q.E.D.


Perfect Numbers:

  • Perfect Number: a number whose factors add up to the number. No one knows if there are any odd perfect numbers but even perfect numbers are of the form: $2^{n-1}(2^n-1)$ where the second factor is a (Mersenne) prime (see below).
  • Then talked about the formula for the Perfect numbers, then through that, talked about the Mersenne Primes.

Mersenne Primes/Primes in General:

  • Primes of the form $2^n - 1$.
  • Connection between n being prime and $2^n - 1$ being prime? A formula? Can this be proved?
  • Difference between largest Mersenne, largest known Mersenne.
  • $2^n-1$ can only have the chance to be primes when $n$ is is prime.
  • Euclid proved the infinitude of Prime numbers.
  • Not known if there is infinite Mersenne primes.
  • Computationally intensive. 4 days to test a smaller Mersenne may be considered to be quick.
  • Great Internet Mersenne Prime Search - GIMPS (http://www.mersenne.org/). Dedicate your computer's resources to helping to test for the next Mersenne Prime.

Pythagorean Triples

  • Pythagorean Triples connects to Fermat's Theorem (last).
  • Examples: {3,4,5}, {6,8,10}
  • Primitive vs. "Not Primitive" - Are $x$, $y$ and $z$ relatively prime (no common factors - as in above left - or not (above right)?
  • Is there a formula.

YES
Infinite Series of Pythagorean triples for odd and even numbers:

Odd: 
$x = 2n+1$
 $y = 2n^2+2n$
$z = 2n^2+2n+1$.
 $n=1,2,3,4,5.....$
 $x^2 + y^2 = z^2$.

Even: 
$x = 2n,$
 $y = n^2-1$
 $z = n^2+1$.
 $n=2,3,4,5....$
 $x^2 + y^2 = z^2$. by Mou “夫子之道,忠恕而已矣”

Fermat's Theorems

  • Somewhat based on Pythagoras' Theorem - $x^2 + y^2 = z^2$. Does the same thing hold true if we continue to increase the powers for all three variables?
  • Fermat's Last Theorem (which was more of a conjecture): the equation $x^n+ y^n = z^n$ has no solutions for $n>2$ where $x, y, z$ are not 0. This was later proved by Andrew Wiles.
  • Note that it took until the late twentieth century for Fermat's Last Theorem to be proven, and it has been suggested that not even Fermat knew how to prove this, despite his claims to the contrary.

All the Gauss,and Lobatchevski's work revolutionized the fundaments of Mathematics. Although it was logically admissible, it couldn’t be applied to the physical world. Because of this, the new Geometry was relegated to a pure mind game and to a mathematical deduction.
-Rachel By-Chaparro


Contributors: Lin, Angela, Mingzhi, Rob, Mou
Formatting: Rob


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