nyuuzyou/wikis huggingface dataset:
Euler's Number is Transcendental/Proof 1

== Theorem ==
== Proof ==
 there exist integers $a_0, \ldots, a_n$ with $a_0 \ne 0$ such that:
:$(1): \quad a_n e^n + a_{n - 1} e^{n - 1} + \cdots + a_0 = 0$
Define $M$, $M_1, \ldots, M_n$ and $\epsilon_1, \ldots, \epsilon_n$ as follows:
where $p$ is a prime number with $p > n$ and $p > \size {a_0}$.
The expression $\sqbrk {\paren {x - 1} \cdots \paren {x - n} }^p$ is a polynomial of degree $n p$ with integer coefficients.
Hence:
where $C_\alpha$ are integers and $C_0 = \pm \paren {n!}^p$.
For $\alpha = 0$ we have:
:$C_0 \dfrac {\paren {p - 1}!} {\paren {p - 1}!} = \pm \paren {n!}^p$
Since $p > n$, it follows from Prime iff Coprime to all Smaller Positive Integers and Euclid's Lemma that this term is not divisible by $p$.
For $\alpha \ge 1$ we have:
:$C_\alpha \dfrac {\paren {p - 1 + \alpha}!} {\paren {p - 1}!} = C_\alpha \paren {p - 1 + \alpha} \paren {p - 2 + \alpha} \cdots p$
which is clearly divisible by $p$.
It follows from Common Divisor Divides Difference that $M$ is an integer not divisible by $p$.
We also have:
The expression $\sqbrk {\paren {x + k - 1} \cdots \paren {x + k - n} }$ is divisible by $x$.
So $\paren {x + k}^{p - 1} \sqbrk {\paren {x + k - 1} \cdots \paren {x + k - n} }^p$ is a polynomial of degree at least $p$ with integer coefficients.
Hence:
where $D_\alpha$ are integers.
Since this sum begins with $\alpha = 1$, each term is divisible by $p$. 
Thus each $M_k$ is an integer divisible by $p$.
By the above definitions we have:
:$e^k = \dfrac {M_k + \epsilon_k} M$
Substituting this into $(1)$ and multiplying by $M$ we obtain:
:$\paren {a_0 M + a_1 M_1 + \cdots + a_n M_n} + \paren {a_1 \epsilon_1 + \cdots + a_n \epsilon_n} = 0$
Since $p > \size {a_0}$, it follows from Prime iff Coprime to all Smaller Positive Integers that $p$ does not divide $a_0$.
So by Euclid's Lemma, $a_0 M$ is not divisible by $p$.
Since each $M_k$ is divisible by $p$, it follows from Common Divisor Divides Difference that $a_0 M + a_1 M_1 + \cdots + a_n M_n$ is not divisible by $p$.
Therefore $a_0 M + a_1 M_1 + \cdots + a_n M_n$ is a non-zero integer.
We also have:
Let $A$ be the maximum value of $\size {\paren {x - 1} \cdots \paren {x - n} }$ for $x$ in the interval $\closedint 0 n$.
Then:
By Power over Factorial:
:$\ds \lim_{p \mathop \to \infty} \frac {e^n \paren {n A}^p} {p!} = 0$
So $\size {\epsilon_k}$, and therefore $\size {a_1 \epsilon_1 + \cdots + a_n \epsilon_n}$, can be made arbitrarily small by choosing $p$ sufficiently large. 
It follows that $\paren {a_0 M + a_1 M_1 + \cdots + a_n M_n} + \paren {a_1 \epsilon_1 + \cdots + a_n \epsilon_n}$ is non-zero.
This is a contradiction, so $e$ must be transcendental.
== Historical Note ==
== Sources ==
* 
* 
* : Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental
Category:Euler's Number is Transcendental
Euler's Number to Power of Itself

== Example of Power to Real Number ==
Euler's Number $e$ to the power of itself is approximately equal to:
:$e^e \approx 15 \cdotp 15426 \, 22414 \, 79264 \, 18976 \, 0430 \ldots$
== Sources ==
* : Table $1.1$. Mathematical Constants
* : $\S 1$: Special Constants: $1.12$
Category:Examples of Real Powers
Category:Euler's Number
Euler's Number to Power of its Negative

== Example of Power to Real Number ==
Euler's Number $e$ to the power of its negative is approximately equal to:
:$e^{-e} \approx 0 \cdotp 06598 \, 80358 \, 45312 \ldots$
== Also see ==
* Euler's Number to Power of its Reciprocal
== Sources ==
* : Table $1.1$. Mathematical Constants
* : $0,06598803584 \ldots$
* : $1 \cdotp 444 \, 667 \, 861 \ldots$
* : $1 \cdotp 44466 \, 7861 \ldots$
Category:Examples of Real Powers
Category:Euler's Number
Euler's Number to Power of its Reciprocal

== Example of Power to Real Number ==
Euler's Number $e$ to the power of its reciprocal is approximately equal to:
:$e^{1/e} \approx 1 \cdotp 44466 \, 78610 \, 09766 \, 13365 \, 83 \ldots$
== Also see ==
* Euler's Number to Power of its Negative
== Sources ==
* : $1 \cdotp 444 \, 667 \, 861 \ldots$
* : $1 \cdotp 44466 \, 7861 \ldots$
Category:Examples of Reciprocals
Category:Examples of Real Powers
Category:Euler's Number
Euler's Number to Rational Power permits Unique Continuous Extension

== Theorem ==
Let $e$ be Euler's number.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$f \left({q}\right) = e^q$
where $e^q$ denotes $e$ to the power of $q$.
Then there exists a unique continuous extension of $f$ to $\R$.
== Proof ==
Since $e > 0$, we may apply Power Function to Rational Power permits Unique Continuous Extension.
Hence the result.
Category:Exponential Function
Euler's Pentagonal Numbers Theorem

== Theorem ==
Consider the infinite product:
:$\ds P = \prod_{n \mathop \in \Z_{>0} } \paren {1 - x^n}$
Then $P$ can be expressed as:
:$\ds P = \sum_{n \mathop \in \Z_{>0} } \paren {-1}^{\ceiling {n / 2} } x^{GP_n}$
where:
:$\ceiling {n / 2}$ denotes the ceiling of $n / 2$
:$GP_n$ denotes the $n$th generalized pentagonal number.
That is:
:$P = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + \cdots$
=== Corollary 1 ===
=== Corollary 2 ===
== Proof ==
== Historical Note ==
== Sources ==
* : $22$
* 
* : $22$
Category:Number Theory
Category:Generalized Pentagonal Numbers
Category:Euler's Pentagonal Numbers Theorem
Euler's Pentagonal Numbers Theorem/Corollary 1

== Theorem ==
Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $\map {\sigma_1} n$ denote the divisor sum of $n$.
Then:
:$\map {\sigma_1} n = \ds \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map {\sigma_1} {n - GP_k} + n \sqbrk {\exists k \in \Z: GP_k = n}$
where:
:$\ceiling {k / 2}$ denotes the ceiling of $k / 2$
:$GP_n$ denotes the $n$th generalized pentagonal number
:$\sqbrk {\exists k \in \Z: GP_k = n}$ is Iverson's convention.
== Proof ==
== Examples ==
== Historical Note ==
== Sources ==
* : $22$
* : $22$
Category:Generalized Pentagonal Numbers
Category:Divisor Sum Function
Category:Euler's Pentagonal Numbers Theorem
Euler's Pentagonal Numbers Theorem/Corollary 1/Examples/12

== Example of Euler's Pentagonal Numbers Theorem: Corollary 1 ==
:$\map {\sigma_1} {12} = \map {\sigma_1} {11} + \map {\sigma_1} {10} - \map {\sigma_1} 7 - \map {\sigma_1} 5 + 12$
== Proof ==
== Sources ==
* : $22$
* : $22$
Category:Euler's Pentagonal Numbers Theorem
Category:12
Euler's Pentagonal Numbers Theorem/Corollary 1/Historical Note

== Historical Note on Euler's Pentagonal Numbers Theorem: Corollary 1 ==
 had noticed, but had not formally proved, what is now known as Euler's Pentagonal Numbers Theorem.
However, he was so convinced of its truth that he used it to prove this and other results.
Hence he demonstrated the interesting fact that, in order to calculate the sum of the divisors of a number, it is necessary merely to know the sum of the divisors of the relevant smaller numbers, but not what those divisors themselves actually are.
== Sources ==
* : $22$
* : $22$
Category:Historical Notes
Euler's Pentagonal Numbers Theorem/Corollary 2

== Theorem ==
Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $\map p n$ denote the number of partitions on $n$.
Then:
:$\map p n = \ds \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map p {n - GP_k} + \sqbrk {\exists k \in \Z: GP_k = n}$
where:
:$\ceiling {k / 2}$ denotes the ceiling of $k / 2$
:$GP_n$ denotes the $n$th generalized pentagonal number
:$\sqbrk {\exists k \in \Z: GP_k = n}$ is Iverson's convention.
== Proof ==
== Historical Note ==
== Sources ==
* : $22$
* : $22$
Category:Generalized Pentagonal Numbers
Category:Partition Theory
Category:Euler's Pentagonal Numbers Theorem
Euler's Pentagonal Numbers Theorem/Corollary 2/Historical Note

== Historical Note on Euler's Pentagonal Numbers Theorem: Corollary 2 ==
 had noticed, but had not formally proved, the result now known as Euler's Pentagonal Numbers Theorem.
However, he was so convinced of its truth that he used it to prove this and other results.
== Sources ==
* : $22$
* : $22$
Category:Historical Notes
Euler's Pentagonal Numbers Theorem/Historical Note

== Historical Note on Euler's Pentagonal Numbers Theorem ==
This result was first noticed by , who noticed the pattern on starting to multiply out the given infinite product, but he was initially unable to prove it except by a process of informal induction.
Despite this inability, he was so certain of the pattern that he used it as the basis of the development of further theorems.
== Sources ==
* : $22$
* : $22$
Category:Historical Notes
Euler's Reflection Formula

== Theorem ==
Let $\Gamma$ denote the gamma function.
Then:
:$\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$
=== Corollary 1 ===
=== Corollary 2 ===
== Proof ==
We have the Weierstrass products:
:$\ds \map \sin {\pi z} = \pi z \prod_{n \mathop \ne 0} \paren {1 - \frac z n} \map \exp {\frac z n}$
From the Weierstrass form of the Gamma function:
:$\ds \frac 1 {\map \Gamma z} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {1 + \frac z n} \map \exp {-\frac z n}$
from which:
whence:
== Also see ==
* Digamma Reflection Formula
* Polygamma Reflection Formula
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Special Functions: $\text {I}$. The Gamma function: $3$
* : $\S 16$: Relationships Among Gamma Functions: $16.8$
* : $\S 1.2.5$: Permutations and Factorials: Exercise $23$
* : $\S 25$: The Gamma Function: Relationships Among Gamma Functions: $25.8.$
Category:Euler's Reflection Formula
Category:Gamma Function
Category:Reflection Formulas
Euler's Reflection Formula/Corollary 1

== Corollary to Euler's Reflection Formula ==
Let $\Gamma$ denote the gamma function.
Then:
:$\forall z \notin \Z: \paren {-z}! \, \map \Gamma z = \dfrac \pi {\map \sin {\pi z} }$
== Proof ==
== Sources ==
* : $\S 1.2.5$: Permutations and Factorials
Category:Euler's Reflection Formula
Category:Gamma Function
Euler's Reflection Formula/Corollary 2

== Corollary to Euler's Reflection Formula ==
Let $\Gamma$ denote the gamma function.
Then:
:$\forall \paren {\dfrac 1 2 - z} \notin \Z: \map \Gamma {\dfrac 1 2 - z} \map \Gamma {\dfrac 1 2 + z} = \dfrac \pi {\map \cos {\pi z} }$
== Proof ==
Category:Euler's Reflection Formula
Category:Gamma Function
Euler's Secant Identity

== Theorem ==
:$\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$
where:
:$z \in \C$ is a complex number
:$\sec z$ denotes the secant function
:$i$ denotes the imaginary unit: $i^2 = -1$
== Proof ==
== Also see ==
* Euler's Sine Identity
* Euler's Cosine Identity
* Euler's Tangent Identity
* Euler's Cotangent Identity
* Euler's Cosecant Identity
* Arcsecant Logarithmic Formulation
== Sources ==
* : $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.21$
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
Category:Euler's Identities
Category:Secant Function
Euler's Sine Identity

== Theorem ==
For any complex number $z$:
:$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$
where:
:$\exp z$ denotes the exponential function
:$\sin z$ denotes the complex sine function
:$i$ denotes the inaginary unit.
=== Real Domain ===
This result is often presented and proved separately for arguments in the real domain:
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also presented as ==
== Also see ==
* Euler's Cosine Identity
* Euler's Tangent Identity
* Euler's Cotangent Identity
* Euler's Secant Identity
* Euler's Cosecant Identity
* Arcsine Logarithmic Formulation
== Sources ==
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
* 
* 
Category:Euler's Sine Identity
Category:Euler's Identities
Category:Sine Function
Euler's Sine Identity/Also presented as

== Euler's Sine Identity: Also presented as == 
Euler's Sine Identity can also be presented in the form:
:$\sin z = \dfrac 1 2 i \paren {e^{-i z} - e^{i z} }$
which follows from the given form by multiplying top and bottom by $-i$.
Category:Euler's Sine Identity
Euler's Sine Identity/Real Domain

== Theorem ==
For any real number $x \in \R$:
:$\sin x = \dfrac {e^{i x} - e^{-i x} } {2 i}$
where:
:$e^{i x}$ denotes the exponential function
:$\sin x$ denotes the real sine function
:$i$ denotes the inaginary unit.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also presented as ==
== Also see ==
* Euler's Cosine Identity
* Euler's Tangent Identity
* Euler's Cotangent Identity
* Euler's Secant Identity
* Euler's Cosecant Identity
* Arcsine Logarithmic Formulation
== Sources ==
* : $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.17$
Category:Euler's Sine Identity
Category:Sine Function
Euler's Sum of Powers Conjecture

== Famous False Conjecture ==
No $n$th power can be the sum of fewer than $n$ $n$th powers.
== Refutation ==
:$144^5 = 27^5 + 84^5 + 110^5 + 133^5$
== Historical Note ==
== Sources ==
* : $144$
* : $144$
Category:Number Theory
Category:Famous False Conjectures
Euler's Sum of Powers Conjecture/Historical Note

== Historical Note on Euler's Sum of Powers Conjecture ==
This was proven false by the counterexample found by and in $1966$, when they were using a computer to hunt for $5$th powers which were the sum of $5$ $5$th powers.
In one of the $4$ solutions they found, one of the contributing $5$th powers was $0^5$.
It was at that point it was realised that here was a counterexample to Euler's Sum of Powers Conjecture.
Since then, further counterexamples have been found, for various powers.
== Sources ==
* : $144$
* : $144$
Category:Historical Notes
Euler's Tangent Identity

== Theorem ==
Let $z$ be a complex number.
Let $\tan z$ denote the tangent function and $i$ denote the imaginary unit: $i^2 = -1$.
Then:
=== Formulation 1 ===
=== Formulation 2 ===
=== Formulation 3 ===
== Also see ==
* Euler's Sine Identity
* Euler's Cosine Identity
* Euler's Cotangent Identity
* Euler's Secant Identity
* Euler's Cosecant Identity
* Arctangent Logarithmic Formulation
Category:Euler's Tangent Identity
Category:Euler's Identities
Category:Tangent Function
Euler's Tangent Identity/Formulation 1

== Theorem ==
Let $z$ be a complex number.
Let $\tan z$ denote the tangent function and $i$ denote the imaginary unit: $i^2 = -1$.
Then:
:$\tan z = i \dfrac {1 - e^{2 i z} } {1 + e^{2 i z} }$
== Proof ==
Category:Euler's Tangent Identity
Euler's Tangent Identity/Formulation 2

== Theorem ==
Let $z$ be a complex number.
Let $\tan z$ denote the tangent function and $i$ denote the imaginary unit: $i^2 = -1$.
Then:
:$\tan z = \dfrac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }$
== Proof ==
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$
* : $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.19$
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
Category:Euler's Tangent Identity
Euler's Tangent Identity/Formulation 3

== Theorem ==
Let $z$ be a complex number.
Let $\tan z$ denote the tangent function and $i$ denote the imaginary unit: $i^2 = -1$.
Then:
:$\tan z = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$
== Proof ==
== Sources ==
* : $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.19$
Category:Euler's Tangent Identity
Euler's Theorem (Number Theory)

== Theorem ==
Let $a, m \in \Z$ be coprime integers: $a \perp m$.
Let $\map \phi m$ be the Euler $\phi$ function of $m$.
Then:
:$a^{\map \phi m} \equiv 1 \pmod m$
=== Corollary ===
== Proof ==
Let $\eqclass a m$ denote the residue class modulo $m$ of $a$.
Since $a \perp m$, it follows by Reduced Residue System under Multiplication forms Abelian Group that $\eqclass a m$ belongs to the abelian group $\struct {\Z'_m, \times}$.
Let $k = \order {\eqclass a m}$ where $\order {\, \cdot \,}$ denotes the order of a group element.
By Order of Element Divides Order of Finite Group:
:$k \divides \order {\Z'_m}$
By the definition of the Euler $\phi$ function:
:$\order {\Z'_m} = \map \phi m$
Thus:
== Also presented as ==
== Also see ==
* Fermat's Little Theorem
== Sources ==
* : Chapter $2$: Subgroups and Cosets: $\S 42$
* : $\S 2.3$: Congruences: Exercise $5$
* 
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $28$
* 
* 
* 
* 
Category:Euler's Theorem (Number Theory)
Category:Euler's Theorems
Category:Number Theory
Euler's Theorem (Number Theory)/Corollary 1

== Corollary to Euler's Theorem (Number Theory) ==
Let $p^n$ be a prime power for some prime number $p > 1$.
Let $a$ be an integer not divisible by $p: p \nmid a$.
Then:
:$a^{\paren {p - 1} p^{n - 1} } \equiv 1 \pmod {p^n}$
== Proof ==
We have that Divisor Relation is Transitive.
Since $p \divides p^n$, it follows that:
:$p^n \nmid a$
From Euler's Theorem (Number Theory):
:$a^{\map \phi {p^n} } \equiv 1 \pmod {p^n}$
From Euler Phi Function of Prime Power:
:$\map \phi {p^n} = \paren {p - 1} p^{n - 1}$
Then:
:$a^{\paren {p - 1} p^{n - 1} } \equiv 1 \pmod {p^n}$
Category:Prime Numbers
Category:Euler's Theorem (Number Theory)
Euler's Transformation

== Theorem ==
Let $a, b, c \in \C$.
Let $\size x < 1$
Let $\map \Re c > \map \Re b > 0$.
Then:
:$\ds \map F {a, b; c; x} = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}$
where $\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$.
== Proof ==
First, we observe:
Applying Pfaff's Transformation twice, we obtain:
Therefore, after two applications of Pfaff's Transformation, we have:
:$\map F {a, b; c; x} = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}$
== Also see ==
* Euler's Integral Representation of Hypergeometric Function
* Kummer's Quadratic Transformation
* Pfaff's Transformation
== Sources ==
* : $\S 31$: Hypergeometric Functions: Miscellaneous Properties: $31.17$
* : Chapter $\text {2}$. The Hypergeometric Functions
* 
Category:Gaussian Hypergeometric Function
Euler-Bernoulli Beam Equation

== Theorem ==
:$q = \map {\dfrac {\d^2} {\d x^2} } {E I \dfrac {\d^2 w} {\d x^2} }$
where:
:$q$ is the bending moment
:$E$ is Young's modulus
:$\dfrac {\d^2 w} {\d x^2}$ is the curvature
:$I$ is the moment of inertia of the cross-section about an axis through the center of mass and perpendicular to the plane of the couple.
== Proof ==
== Also known as ==
The Euler-Bernoulli Beam Equation is also known as the Euler-Bernoulli Law.
== Sources ==
* : $\S 3$: Appendix $\text A$: Euler
* 
* : Chapter $\text {A}.21$: Euler ()
Category:Mechanics
Euler-Binet Formula

== Theorem ==
The Fibonacci numbers have a closed-form solution:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n} {\phi - \paren {1 - \phi}}$
where $\phi$ is the golden mean.
Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$
From : $\phi = \dfrac {1 + \sqrt 5} 2$
Therefore, substituting $\sqrt 5 = 2\phi - 1 = \phi - \paren {1 - \phi} = \phi - \hat \phi$, the above can be written as:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\paren {\phi - \hat \phi}}$
=== Corollary 1 ===
=== Corollary 2 ===
=== Negative Index ===
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Also known as ==
== Historical Note ==
== Sources ==
* 
* 
* : $5$
* : Liber Abaci: $88$
* : $5$
Category:Fibonacci Numbers
Category:Golden Mean
Category:Euler-Binet Formula
Category:Closed Forms
Euler-Binet Formula/Corollary 1

== Corollary to Euler-Binet Formula ==
:$F_n = \dfrac {\phi^n} {\sqrt 5}$ rounded to the nearest integer
where:
:$F_n$ denotes the $n$th Fibonacci number
:$\phi$ denotes the golden mean.
== Proof ==
By definition of $n$th Fibonacci number, $F_n$ is an integer.
From Euler-Binet Formula:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5} = \dfrac {\phi^n } {\sqrt 5} - \dfrac {\hat \phi^n} {\sqrt 5}$
But $\size {\dfrac {\hat \phi^n} {\sqrt 5} } < \dfrac 1 2$ for all $n \ge 0$.
Thus $\dfrac {\phi^n } {\sqrt 5}$ differs from $F_n$ by a number less than $\dfrac 1 2$.
Thus the nearest integer to $\dfrac {\phi^n } {\sqrt 5}$ is $F_n$.
== Sources ==
* : $5$
* : $\S 1.2.8$: Fibonacci Numbers: $(15)$
* : $5$
Category:Euler-Binet Formula
Euler-Binet Formula/Corollary 1/Mistake

== Source Work ==
:
:The Dictionary
:: $5$
== Mistake ==
:There is another version of this formula, which is simpler to use in practice. Because
::$\dfrac 1 {\sqrt 5} \paren {\dfrac {1 + \sqrt 5} 2}^n$
is only $0 \cdotp 618 \ldots$ when $n = 1$ and rapidly becomes very small indeed, $F_n$ is actually the nearest integer to
::$\paren {\dfrac {\sqrt 5 - 1} 2}^n$
== Correction ==
The expressions are the wrong way round.
The section ought to read:
:There is another version of this formula, which is simpler to use in practice. Because
::$\paren {\dfrac {\sqrt 5 - 1} 2}^n$
is only $0 \cdotp 618 \ldots$ when $n = 1$ and rapidly becomes very small indeed, $F_n$ is actually the nearest integer to
::$\dfrac 1 {\sqrt 5} \paren {\dfrac {1 + \sqrt 5} 2}^n$
Interestingly and oddly, this is correct in the first edition of of $1986$.
== Sources ==
* : $5$
Category:Mistakes/Curious and Interesting Numbers
Euler-Binet Formula/Corollary 2

== Corollary to Euler-Binet Formula ==
For even $n$:
:$F_n < \dfrac {\phi^n} {\sqrt 5}$
For odd $n$:
:$F_n > \dfrac {\phi^n} {\sqrt 5}$
where:
:$F_n$ denotes the $n$th Fibonacci number
:$\phi$ denotes the golden mean.
== Proof ==
From Euler-Binet Formula:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5} = \dfrac {\phi^n} {\sqrt 5} - \dfrac {\hat \phi^n} {\sqrt 5}$
We have that:
:$\hat \phi = -\dfrac 1 \phi$
and so:
:$\hat \phi^n = \paren {-1}^n \dfrac 1 {\phi^n}$
$\phi > 0$ and so $\dfrac 1 {\phi^n} > 0$ for all $n$.
For even $n$:
: $\paren {-1}^n = 1$
and so:
:$\hat \phi^n > 0$
Thus:
:$\dfrac {\phi^n} {\sqrt 5} - \dfrac {\hat \phi^n} {\sqrt 5} < \dfrac {\phi^n} {\sqrt 5}$
and so $F_n < \dfrac {\phi^n} {\sqrt 5}$.
For odd $n$:
: $\paren {-1}^n = -1$
and so:
:$\hat \phi^n < 0$
Thus:
:$\dfrac {\phi^n} {\sqrt 5} - \dfrac {\hat \phi^n} {\sqrt 5} > \dfrac {\phi^n} {\sqrt 5}$
and so $F_n > \dfrac {\phi^n} {\sqrt 5}$.
== Sources ==
* : $\S 1.2.8$: Fibonacci Numbers: Exercise $10$
Category:Euler-Binet Formula
Euler-Binet Formula/Negative Index

== Theorem ==
Let $n \in \Z_{< 0}$ be a negative integer.
Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers).
Then the Euler-Binet Formula:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5} = \dfrac {\phi^n - \hat \phi^n} {\phi - \hat \phi}$
continues to hold.
In the above:
:$\phi$ is the golden mean
:$\hat \phi = 1 - \phi = -\dfrac 1 \phi$
== Proof ==
Let $n \in \Z_{> 0}$.
Then:
== Sources ==
* : $\S 1.2.8$: Fibonacci Numbers: Exercise $9$
Category:Euler-Binet Formula
Euler-Binet Formula/Proof 1

== Theorem ==
== Proof ==
Proof by induction:
For all $n \in \N$, let $\map P n$ be the proposition:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$
=== Basis for the Induction ===
$\map P 0$ is true, as this just says:
:$\dfrac {\phi^0 - \hat \phi^0} {\sqrt 5} = \dfrac {1 - 1} {\sqrt 5} = 0 = F_0$
$\map P 1$ is the case:
This is our basis for the induction.
=== Induction Hypothesis ===
Now we need to show that, if $\map P j$ is true for all $0 \le j \le k + 1$, then it logically follows that $\map P {k + 2}$ is true.
So this is our induction hypothesis:
:$\forall 0 \le j \le k + 1: F_j = \dfrac {\phi^j - \hat \phi^j} {\sqrt 5}$
Then we need to show:
:$F_{k + 2} = \dfrac {\phi^{k + 2} - \hat \phi^{k + 2} } {\sqrt 5}$
=== Induction Step ===
This is our induction step:
We observe that we have the following two identities:
:$\phi^2 = \paren {\dfrac {1 + \sqrt 5} 2}^2 = \dfrac 1 4 \paren {6 + 2 \sqrt 5} = \dfrac {3 + \sqrt 5} 2 = 1 + \phi$
:$\hat \phi^2 = \paren {\dfrac {1 - \sqrt 5} 2}^2 = \dfrac 1 4 \paren {6 - 2 \sqrt 5} = \dfrac {3 - \sqrt 5} 2 = 1 + \hat \phi$
This can also be deduced from the definition of the golden mean: the fact that $\phi$ and $\hat \phi$ are the solutions to the quadratic equation $x^2 = x + 1$.
Thus:
The result follows by the Second Principle of Mathematical Induction.
Therefore:
:$\forall n \in \N: F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$
== Also known as ==
Category:Euler-Binet Formula