23 Jan 2020 - Pranay Yadav
Infinitude of the Primes
Two notions held most beautiful in mathematics: prime numbers and infinity. Combining the two presents the notion of infinitude of prime numbers. Euclid in 500 BC proved the set of prime numbers, \(P\) to be an \(\infty\) (or unbounded) set - that there is no last prime number.
Challenger-prover paradigm is a game-theoretic way of proving where a challenger picks an \(M\), and the prover finds a prime \(p\) greater than \(M\). More generally, the challenger picks a bound and the prover defeats this bound.
\(example:\ \text{let}\ M=10^{20},\ \text{then}\ \exists p\in P\ \text{s.t.}\ p>M\) Euclid’s proof for the infinitude of primes builds upon the fundamental theorem of arithmetic (FToA), which states that any integer \(N\) can be written as a product of prime numbers. Proving the FToA is non-trivial and involves complex analysis.
\[\text{FToA}:\forall N\in \mathbb N,\ N=\prod_{i=1}^k p_i^{a_i}\quad \text{where}\ p_i\in P,\ a_i\in \mathbb N^+\cup\{0\} \\ \small{^*decomposition\ of\ N\ is\ unique\ up\ to\ a\ permutation}\]Euclid’s proof by contradiction:
\(\text{Assume that } P \text{ is bounded by } p_M \\ P = \{p_1,p_2,\ldots,p_M\} \\ \text{Using FToA, let } x = \prod_{i=1}^M p_i \quad (\text{note that }x > p_M) \\ \text{adding 1 leaves a remainder: } x = \prod_{i=1}^M p_i+1 \quad \text{as }p_i\nmid x \\ \text{thus, either } x\text{ is a prime } \rightarrow \text{ contradicts the original assumption, or} \\ \text{there exists a prime} \notin P \text{ that divides } x \rightarrow \text{ contradicts the bound on }P \\\) Euler’s proof for the infinitude of primes builds on from the following observations about harmonic series.
\[S_a = \sum_{n=0}^{\infty}\frac{1}{2^n} = 1+\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+\ldots=2 \qquad \text{(converges)} \\ S = \sum_{n=1}^{\infty}\frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots=\infty \qquad \text{(diverges, slowly)}\]Similar to the challenger-prover paradigm, if \(T\), a subset of \(S\) representing a lower bound, diverges, then \(S\) also diverges. However, if the lower bound converges then \(S\) may or may not converge, while if an upper bound on \(S\) converges then \(S\) also converges.
\[\text{choose a }T \le S \quad \text{by replacing numbers from } S\ \text{ s.t. }\ T \rightarrow\infty \\ S =1+\frac{1}{2}+(\frac{1}{3}+\frac{1}{4})+(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8})+\ldots \quad\small{(parentheses\ represent\ partial\ sums)}\\ T =1+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8})+\ldots \quad\small{(parentheses\ represent\ partial\ sums)}\\ T =1+\frac{1}{2}+(\frac{1}{2}) +(\frac{1}{2})+\ldots \qquad \text{(diverges, thus S diverges)} \\\]This proof of the divergence of the harmonic series was first proposed by Nicole Oresme in the 14th century. In his elegant proof, Euler showed that a harmonic series of prime numbers diverges:
\[A = \frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\ldots+\frac{1}{p_m} \\ A\rightarrow \infty\implies P\rightarrow \infty \\ \text{where } P = \{p_1, p_2, \ldots,p_m\}\]More on prime numbers
Twin primes are pairs of prime numbers separated by 2 \((p,p+2)\). That there are \(\infty\) of these is the twin prime conjecture, a special case of Polignac’s conjecture which states that there are \(\infty\) consecutive primes separated by a gap $$k$.
\(\rightarrow\) Prove that the number in between twin primes \((p+1)\) is divisible by 6
The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of 2 primes. Example: \(12=5+7\)
Parallel postulate
Euclid’s fifth postulate is taken as as an axiom if there is exactly one possible line parallel to a given line passing through a point. Depending on the choice of axiom for the number of possible parallel lines, broadly three systems of geometry result:
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\(0\rightarrow\) Non-Euclidean (spherical/elliptical) geometry where the definition of a line (shortest path between 2 points) is fulfilled by great circles (equators) or geodesics. All geodesics on a sphere intersect, in other words, no geodesics are parallel to each other.
Fun fact: Airplanes travel along geodesics
- \(1\rightarrow\) Euclidean geometry, example \(\mathbb R^2\)
- \(\infty\rightarrow\) Non-Euclidean (hyperbolic) geometry, used to model space-time in theory of relativity.
Infinity
Since \(\mathbb N\) is countably infinite, another set \(A\) can be shown to be countably infinite by demonstrating a one-to-one mapping, \(f\), from \(A\) to \(\mathbb N\). This one-to-one mapping shows identical cardinality. Example: \(f:x\rightarrow \frac{x}{2}\)
Functions and Mappings
Injection (one-to-one): \(f:D\rightarrow R\ \text{where}\ \forall x_1,x_2\in D,\, y_1=f(x_1),\ y_2=f(x_2),\ \text{s.t. }x_1\ne x_2\implies y_1\ne y_2\) Example of one-to-one but not onto: \(\mathbb N \rightarrow \mathbb N :\) \(x\rightarrow 2x\)
Surjection (onto):
\(f:D\rightarrow R\ \text{where}\ \forall y\in R,\, \exists x\in D,\ \text{s.t. }y=f(x)\) Example of onto but not one-to-one: \(\mathbb R \rightarrow \mathbb R^+\cup\{0\}:\) \(x\rightarrow |x|\)
Bijection (one-to-one & onto): Both Injective and Surjective, are invertible.
Example: \(\mathbb N \rightarrow \mathbb N :\) \(f(x)\rightarrow x\) The set of rationals, \(\mathbb Q = \{\dfrac{p}{q},\ \ p,q\in (\mathbb Z , q\ne 0\}\) can be thus shown to be countably infinite.
Properties of Numbers
As we expand the space of numbers from integers and natural numbers to real and complex numbers, we go from countably infinite to uncountably infinite. There is a loss of certain properties with each level of construction in the hierarchy of numbers:
- \(\mathbb N , \mathbb Z \ \small{countably_{\infty}}\ \rightarrow\) nextness or adjacency of numbers
- \(\mathbb Q \ \small{countably_{\infty}}\rightarrow\) dense, no nextness, between any 2 elements, another element can be found.
- \(\mathbb R \ \small{uncountably_{\infty}}\rightarrow\) ordering exists, but denumerable (\(\mathbb R \not\rightarrow \mathbb N\))
- \(\mathbb C \ \small{uncountably_{\infty}}\rightarrow\) no ordering exists
Certain properties are also gained with each level:
- \(\mathbb N , \mathbb Z \rightarrow\) possible solutions of Diophantine equations
- \(\mathbb Q \rightarrow\) solutions of all equations of the form \(ax+b=0,\ a,b\in \mathbb Z\)
- \(\mathbb R \rightarrow\) Can be divided into rationals and irrationals. Another way is either algebraic (\(x\in \mathbb R\) s.t. \(x\) is a solution to algebraic equations) or transcendental (\(\subset\) irrationals, not solutions to any algebraic equations)
- \(\mathbb C \rightarrow\) Always contains all possible solutions to \(a_1x^n+a_2x^{n-1}+\ldots+a_n=0,\ a_i\in \mathbb R , \mathbb C\) (Fundamental Theorem of Algebra)
Further explorations:
- Proofs from the Book by Aigner & Ziegler (link)
- The Man Who Loved Only Numbers by Paul Hoffman on Paul Erdős (link) (lecture)
- N is a number - a documentary on Paul Erdős (playlist)
- Number Theory in Science and Communication by Manfred Schroeder (link)
- Classification of all the numbers by Matt Parker on Numberphile (link)