nyuuzyou/wikis huggingface dataset:
Trichotomy Law (Ordering)

== Theorem ==
Let $\struct {S, \preceq}$ be an ordered set.
Then $\preceq$ is a total ordering :
:$\forall a, b \in S: \paren {a \prec b} \lor \paren {a = b} \lor \paren {a \succ b}$
That is, every element either strictly precedes, is the same as, or strictly succeeds, every other element.
In other words, $\prec$ is a trichotomy.
== Proof ==
== Also known as ==
== Sources ==
* : Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
* : Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order
Category:Trichotomies
Category:Trichotomy Law
Category:Total Orderings
Category:Named Theorems
Trichotomy Law for Natural Numbers

== Theorem ==
Let $\omega$ be the set of natural numbers defined as the von Neumann construction.
Let $m, n \in \omega$.
Then one of the following cases holds:
:$m \in n$
:$m = n$
:$n \in m$
== Proof ==
By definition of the ordering on von Neumann construction:
:$m \le n \iff m \subseteq n$
From Natural Number m is Less than n iff m is an Element of n, we have:
:$m < n \iff m \in n$
Hence the theorem is equivalent to the statement that for every $m, n \in \omega$, one of the following holds:
:$m \subsetneq n$
:$m = n$
:$n \subsetneq m$
The result follows from Natural Numbers are Comparable: Strong Result.
== Also known as ==
== Sources ==
* : Chapter $3$: The Natural Numbers: $\S 5$ Applications to natural numbers: Theorem $5.7$
Category:Ordering on Natural Numbers
Trichotomy Law for Real Numbers

== Theorem ==
The real numbers obey the trichotomy law.
That is, $\forall a, b \in \R$, exactly one of the following holds:
== Proof 1 ==
== Proof 2 ==
== Also known as ==
== Sources ==
* : $\S 1.1$. Number Systems
* : Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-2}$: Inequalities
* : $\S 1$: Real Numbers: $\S 1.4$: Inequalities
* : $(1)$ Trichotomy law
* 
* : $(1)$ Trichotomy law
* 
Category:Trichotomy Law for Real Numbers
Category:Real Numbers
Category:Inequalities
Category:Trichotomy Law
Trichotomy is Antireflexive

== Theorem ==
Let $\RR$ be a trichotomy.
Then $\RR$ is an antireflexive relation.
== Proof ==
Let $\RR$ be a trichotomy on a set $S$.
Let $x \in S$.
By definition of a trichotomy, for all $a, b \in S$, either:
:$a \mathrel \RR b$
:$a = b$
:$b \mathrel \RR a$
As $x = x$ it follows directly that $x \not < x$.
Hence the result by definition of antireflexive relation.
Category:Antireflexive Relations
Category:Trichotomies
Trigonometric Addition Formulas

== Theorem ==
=== Sine of Sum ===
=== Cosine of Sum ===
=== Tangent of Sum ===
== Also defined as ==
== Also known as ==
== Also see ==
* Trigonometric Subtraction Formulas
* Addition Formulas for Hyperbolic Functions
* Subtraction Formulas for Hyperbolic Functions
== Sources ==
* 
* 
* 
Category:Trigonometric Addition Formulas
Category:Compound Angle Formulas
Category:Trigonometric Identities
Trigonometric Addition Formulas/Also defined as

== Trigonometric Addition Formulas: Also defined as ==
Some sources include the Trigonometric Subtraction Formulas in with the Trigonometric Addition Formulas.
Other sources lump the Trigonometric Addition Formulas and Trigonometric Subtraction Formulas together and call them all Compound Angle Formulas.
== Sources ==
* 
Category:Trigonometric Addition Formulas
Trigonometric Addition Formulas/Also known as

== Trigonometric Addition Formulas: Also known as ==
The trigonometric addition formulas are also known as the trigonometric addition formulae.
They are also known as:
* the addition formulas or addition formulae
* Ptolemy's formulas or Ptolemy's formulae, for .
* Simpson's formulas or Simpson's formulae, for , although this name has been given to various sets of trigonometric identities, which can cause confusion.
Some sources use the term trigonometric addition formulas as a synonym for compound angle formulas.
== Sources ==
* 
* 
* 
* 
* 
* 
Category:Trigonometric Addition Formulas
Trigonometric Functions in terms of each other

== Theorem ==
=== Sine in terms of Cosine ===
=== Sine in terms of Tangent ===
=== Sine in terms of Cotangent ===
=== Sine in terms of Secant ===
=== Sine is Reciprocal of Cosecant ===
=== Cosine in terms of Sine ===
=== Cosine in terms of Tangent ===
=== Cosine in terms of Cotangent ===
=== Cosine is Reciprocal of Secant ===
=== Cosine in terms of Cosecant ===
=== Tangent in terms of Sine ===
=== Tangent in terms of Cosine ===
=== Tangent is Reciprocal of Cotangent ===
=== Tangent in terms of Secant ===
=== Tangent in terms of Cosecant ===
=== Cotangent in terms of Sine ===
=== Cotangent in terms of Cosine ===
=== Cotangent is Reciprocal of Tangent ===
=== Cotangent in terms of Secant ===
=== Cotangent in terms of Cosecant ===
=== Secant in terms of Sine ===
=== Secant is Reciprocal of Cosine ===
=== Secant in terms of Tangent ===
=== Secant in terms of Cotangent ===
=== Secant in terms of Cosecant ===
=== Cosecant is Reciprocal of Sine ===
=== Cosecant in terms of Cosine ===
=== Cosecant in terms of Tangent ===
=== Cosecant in terms of Cotangent ===
=== Cosecant in terms of Secant ===
Category:Sine Function
Category:Cosine Function
Category:Tangent Function
Category:Cosecant Function
Category:Secant Function
Category:Cotangent Function
Category:Trigonometric Functions
Trigonometric Functions of Angle plus Full Angle

== Theorem ==
=== Sine of Angle plus Full Angle ===
=== Cosine of Angle plus Full Angle ===
=== Tangent of Angle plus Full Angle ===
=== Cotangent of Angle plus Full Angle ===
=== Secant of Angle plus Full Angle ===
=== Cosecant of Angle plus Full Angle ===
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Trigonometric Identities
Trigonometric Functions of Angle plus Right Angle

== Theorem ==
=== Sine of Angle plus Right Angle ===
=== Cosine of Angle plus Right Angle ===
=== Tangent of Angle plus Right Angle ===
=== Cotangent of Angle plus Right Angle ===
=== Secant of Angle plus Right Angle ===
=== Cosecant of Angle plus Right Angle ===
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Trigonometric Identities
Trigonometric Functions of Angle plus Straight Angle

== Theorem ==
=== Sine of Angle plus Straight Angle ===
=== Cosine of Angle plus Straight Angle ===
=== Tangent of Angle plus Straight Angle ===
=== Cotangent of Angle plus Straight Angle ===
=== Secant of Angle plus Straight Angle ===
=== Cosecant of Angle plus Straight Angle ===
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Trigonometric Identities
Trigonometric Functions of Angle plus Three Right Angles

== Theorem ==
=== Sine of Angle plus Three Right Angles ===
=== Cosine of Angle plus Three Right Angles ===
=== Tangent of Angle plus Three Right Angles ===
=== Cotangent of Angle plus Three Right Angles ===
=== Secant of Angle plus Three Right Angles ===
=== Cosecant of Angle plus Three Right Angles ===
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Trigonometric Identities
Trigonometric Functions of Complementary Angles

== Theorem ==
=== Sine of Complement equals Cosine ===
=== Cosine of Complement equals Sine ===
=== Tangent of Complement equals Cotangent ===
=== Cotangent of Complement equals Tangent ===
=== Secant of Complement equals Cosecant ===
=== Cosecant of Complement equals Secant ===
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Trigonometric Identities
Category:Complementary Angles
Trigonometric Functions of Conjugate Angles

== Theorem ==
=== Sine of Conjugate Angle ===
=== Cosine of Conjugate Angle ===
=== Tangent of Conjugate Angle ===
=== Cotangent of Conjugate Angle ===
=== Secant of Conjugate Angle ===
=== Cosecant of Conjugate Angle ===
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Trigonometric Identities
Category:Conjugate Angles
Trigonometric Functions of Negative Angle

== Theorem ==
=== Sine Function is Odd ===
=== Cosine Function is Even ===
=== Tangent Function is Odd ===
=== Cotangent Function is Odd ===
=== Secant Function is Even ===
=== Cosecant Function is Odd ===
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Trigonometric Identities
Trigonometric Functions of Supplementary Angles

== Theorem ==
=== Sine of Supplementary Angle ===
=== Cosine of Supplementary Angle ===
=== Tangent of Supplementary Angle ===
=== Cotangent of Supplementary Angle ===
=== Secant of Supplementary Angle ===
=== Cosecant of Supplementary Angle ===
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Trigonometric Identities
Category:Supplementary Angles
Trigonometric Functions of Three Right Angles less Angle

== Theorem ==
=== Sine of Three Right Angles less Angle ===
=== Cosine of Three Right Angles less Angle ===
=== Tangent of Three Right Angles less Angle ===
=== Cotangent of Three Right Angles less Angle ===
=== Secant of Three Right Angles less Angle ===
=== Cosecant of Three Right Angles less Angle ===
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Trigonometric Identities
Trigonometric Identities

== Compound Angle Formulas ==
== Sum of Squares of Sine and Cosine ==
=== Corollary 1 ===
=== Corollary 2 ===
== Double Angle Formulas ==
== Half Angle Formulas ==
== Simpson's Formulas ==
== Prosthaphaeresis Formulas ==
== Sum Formulas for Sine and Cosine ==
== Power Reduction Formulas ==
== Minor Identities ==
Category:Trigonometry
Category:Trigonometric Identities
Trigonometric Series is Convergent if Sum of Absolute Values of Coefficients is Convergent

== Theorem ==
Let $\map S x$ be a trigonometric series:
:$\map S x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
Let the series:
:$\ds \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$
be convergent.
Then $\map S x$ is a convergent series for each $x \in \R$.
== Proof ==
For all $n \in \N_{\ge 1}$ and $x \in \R$, we have:
We have , the series $\ds \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$ is convergent.
By the Comparison Test, it follows that:
:$\ds \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$
is absolutely convergent for all $x \in \R$.
From Absolutely Convergent Real Series is Convergent, it follows that $\map S x$ is convergent for all $x \in \R$.
== Sources ==
* : Chapter One: $\S 1$. Trigonometrical Series
Category:Trigonometric Series
Category:Convergence
Trigonometric Subtraction Formulas

== Theorem ==
=== Sine of Difference ===
=== Cosine of Difference ===
=== Tangent of Difference ===
== Also known as ==
== Also see ==
* Addition Formulas
* Addition Formulas for Hyperbolic Functions
* Subtraction Formulas for Hyperbolic Functions
== Sources ==
* 
* 
* 
* 
Category:Trigonometric Subtraction Formulas
Category:Compound Angle Formulas
Category:Trigonometric Identities
Trigonometric Subtraction Formulas/Also known as

== Trigonometric Subtraction Formulas: Also known as ==
The trigonometric subtraction formulas are also known as the trigonometric subtraction formulae.
They are also known just as the subtraction formulas or subtraction formulae.
They are often seen categorised along with the trigonometric addition formulas as the compound angle formulas.
== Sources ==
* 
* 
* 
* 
Category:Trigonometric Subtraction Formulas
Trimorphic Number is not necessarily Automorphic

== Theorem ==
Let $n \in \Z_{>0}$ be a trimorphic number.
Then it is not necessarily the case that $n$ is also an automorphic number.
== Proof ==
Take as an example $n = 49$.
We have that:
:$49^3 = 117 \, 6 \mathbf{49}$
demonstrating that $49$ is trimorphic.
However, we also have that:
:$49^2 = 2401$
demonstrating that $49$ is not automorphic.
== Sources ==
* : $49$
* : $49$
Category:Trimorphic Numbers
Category:Automorphic Numbers
Triperfect Number/Examples/1,476,304,896

== Example of Triperfect Number ==
$1 \, 476 \, 304 \, 896$ is triperfect:
:$\map {\sigma_1} {1 \, 476 \, 304 \, 896} = 4 \, 428 \, 914 \, 688 = 3 \times 1 \, 476 \, 304 \, 896$
== Proof ==
From :
:$\map {\sigma_1} {1 \, 476 \, 304 \, 896} = 4 \, 428 \, 914 \, 688$
Hence the result.
Category:Triperfect Numbers
Category:1,476,304,896
Triperfect Number/Examples/120

== Example of Triperfect Number ==
$120$ is triperfect:
:$\map {\sigma_1} {120} = 360 = 3 \times 120$
== Proof ==
From :
:$\map {\sigma_1} {120} = 360$
Hence the result.
== Sources ==
* : Preface
Category:Triperfect Numbers
Category:120
Triperfect Number/Examples/459,818,240

== Example of Triperfect Number ==
$459 \, 818 \, 240$ is triperfect:
:$\map {\sigma_1} {459 \, 818 \, 240} = 1 \, 379 \, 454 \, 720 = 3 \times 459 \, 818 \, 240$
== Proof ==
From :
:$\map {\sigma_1} {459 \, 818 \, 240} = 1 \, 379 \, 454 \, 720$
Hence the result.
Category:Triperfect Numbers
Category:459,818,240
Triperfect Number/Examples/51,001,180,160

== Example of Triperfect Number ==
$51 \, 001 \, 180 \, 160$ is triperfect:
:$\map {\sigma_1} {51 \, 001 \, 180 \, 160} = 153 \, 003 \, 540 \, 480 = 3 \times 51 \, 001 \, 180 \, 160$
== Proof ==
From :
:$\map {\sigma_1} {51 \, 001 \, 180 \, 160} = 153 \, 003 \, 540 \, 480$
Hence the result.
Category:Triperfect Numbers
Category:51,001,180,160
Triperfect Number/Examples/523,776

== Example of Triperfect Number ==
$523 \, 776$ is triperfect:
:$\map {\sigma_1} {523 \, 776} = 1 \, 571 \, 328 = 3 \times 523 \, 776$
== Proof ==
From :
:$\map {\sigma_1} {523 \, 776} = 1 \, 571 \, 328$
Hence the result.
== Sources ==
* : $523,776$
* : $523,776$
Category:Triperfect Numbers
Category:523,776
Triperfect Number/Examples/672

== Example of Triperfect Number ==
$672$ is triperfect:
:$\map {\sigma_1} {672} = 2016 = 3 \times 672$
== Proof ==
From :
:$\map {\sigma_1} {672} = 2016$
Hence the result.
== Sources ==
* : Preface
Category:Triperfect Numbers
Category:672
Triple Angle Formulas

== Theorem ==
=== Triple Angle Formula for Sine ===
=== Triple Angle Formula for Cosine ===
=== Triple Angle Formula for Tangent ===
where $\sin, \cos, \tan$ denote sine, cosine and tangent respectively.
=== Triple Angle Formula for Hyperbolic Sine ===
=== Triple Angle Formula for Hyperbolic Cosine ===
=== Triple Angle Formula for Hyperbolic Tangent ===
where $\sinh, \cosh, \tanh$ denote hyperbolic sine, hyperbolic cosine and hyperbolic tangent respectively.
== Also see ==
* Double Angle Formulas
* Quadruple Angle Formulas
Category:Triple Angle Formulas
Category:Multiple Angle Formulas
Category:Trigonometric Identities
Category:Hyperbolic Functions
Triple Angle Formulas/Cosine

== Theorem ==
:$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$
where $\cos$ denotes cosine.
=== Example: $2 \cos 3 \theta + 1$ ===
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.45$
* : $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $90 \ \text{(b)}$
* 
* 
Category:Triple Angle Formula for Cosine
Category:Triple Angle Formulas
Category:Cosine Function
Triple Angle Formulas/Cosine/2 cos 3 theta + 1

== Theorem ==
:$2 \cos 3 \theta + 1 = \paren {\cos \theta - \cos \dfrac {2 \pi} 9} \paren {\cos \theta - \cos \dfrac {4 \pi} 9} \paren {\cos \theta - \cos \dfrac {8 \pi} 9}$
== Proof ==
Setting $z = e^{i \theta}$:
== Sources ==
* : $\S 3$. Roots of Unity: Example $6$: $(3.15)$
Category:Triple Angle Formula for Cosine
Category:Cosine Function
Triple Angle Formulas/Cosine/2 cos 3 theta + 1/Mistake

== Source Work ==
:
: Chapter $3$: Roots of Unity: 
:: Example $6$
This mistake can be seen in the $1960$ edition as published by Routledge & Kegan Paul.
== Mistake ==
:Show that
::$z^6 + z^3 + 1 = \paren {z^2 - 2 z \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {8 \pi} 9 + 1} \qquad (3.14)$
:and deduce that
::$2 \cos 3 \theta + 1 = \paren {\cos \theta - \cos \dfrac {2 \pi} 9} \paren {\cos \theta - \cos \dfrac {4 \pi} 9} \paren {\cos \theta - \cos \dfrac {8 \pi} 9} \qquad (3.15)$
:Use $(3.13)$ when $n = 9$ and observe that the factor that corresponds to $r = 3$, is $z^2 - 2 z \cos \dfrac {2 \pi} 3 + 1 = z^2 + z + 1$, the remaining three quadratic factors being as on the of $(3.14)$. This expression is therefore equal to $\paren {z^9 - 1} / \paren {z - 1} \paren {z^2 + z + 1} = \paren {z^9 - 1} / \paren {z^3 - 1} = z^6 + z^3 + 1$, which proves $(3.14)$. Next, divide $(3.14)$ throughout by $z^3$ and then put $z = e^i$. With this value of $z$, $z + z^{-1} = 2 \cos \theta$, $z^3 + z^{-3} = 2 \cos 3 \theta$, and $(3.15)$ is an immediate consequence.
== Correction ==
The substitution $z = e^i$ should instead be $z = e^{i \theta}$.
The full derivation of this exercise is given in Complex Algebra Examples: $z^6 + z^3 + 1$ and Triple Angle Formula for Cosine: $2 \cos 3 \theta + 1$.
== Sources ==
* : $\S 3$. Roots of Unity: Example $6$: $(3.15)$
Category:Mistakes/Complex Numbers/Walter Ledermann
Triple Angle Formulas/Hyperbolic Cosine

== Theorem ==
: $\cosh 3 x = 4 \cosh^3 x - 3 \cosh x$
where $\cosh$ denotes hyperbolic cosine.
== Proof ==
== Also see ==
* Triple Angle Formula for Hyperbolic Sine
* Triple Angle Formula for Hyperbolic Tangent
* Quadruple Angle Formula for Hyperbolic Sine
* Quadruple Angle Formula for Hyperbolic Cosine
* Quadruple Angle Formula for Hyperbolic Tangent
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.31$: Multiple Angle Formulas
Category:Hyperbolic Cosine Function
Category:Triple Angle Formulas
Triple Angle Formulas/Hyperbolic Sine

== Theorem ==
:$\sinh 3 x = 3 \sinh x + 4 \sinh^3 x$
where $\sinh$ denotes hyperbolic sine.
== Proof ==
== Also see ==
* Triple Angle Formula for Hyperbolic Cosine
* Triple Angle Formula for Hyperbolic Tangent
* Quadruple Angle Formula for Hyperbolic Sine
* Quadruple Angle Formula for Hyperbolic Cosine
* Quadruple Angle Formula for Hyperbolic Tangent
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.30$: Multiple Angle Formulas
Category:Hyperbolic Sine Function
Category:Triple Angle Formulas
Triple Angle Formulas/Hyperbolic Tangent

== Theorem ==
:$\tanh {3 x} = \dfrac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}$
where $\tanh$ denotes hyperbolic tangent.
== Proof ==
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.32$: Multiple Angle Formulas
Category:Hyperbolic Tangent Function
Category:Triple Angle Formulas
Triple Angle Formulas/Sine

== Theorem ==
:$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$
where $\sin$ denotes sine.
== Proof 1 ==
== Proof 2 ==
== Historical Note ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.44$
* : $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $90 \ \text{(a)}$
* 
* 
Category:Triple Angle Formula for Sine
Category:Triple Angle Formulas
Category:Sine Function
Triple Angle Formulas/Tangent

== Theorem ==
:$\tan 3 \theta = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$
where $\tan$ denotes tangent.
=== Corollary 1 ===
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.46$
Category:Tangent Function
Category:Triple Angle Formula for Tangent
Category:Triple Angle Formulas
Triple Angle Formulas/Tangent/Corollary 1

== Corollary to Triple Angle Formula for Tangent ==
:$\tan 3 \theta = \tan \theta \dfrac {4 \cos^2 \theta - 1} {4 \cos^2 \theta - 3}$
== Proof ==
Category:Tangent Function
Category:Triple Angle Formula for Tangent
Category:Triple Angle Formulas
Triple Angle Formulas/Tangent/Proof 2

== Theorem ==
== Proof ==
Let $\theta$ be such that $\tan 2 \theta$ is defined.
Then:
Now suppose $\theta$ is such that $\tan 2 \theta$ is not defined.
Then:
:$2 \theta = \dfrac \pi 2 + n \pi$
for some integer $n$.
Hence:
:$\theta = \dfrac \pi 4 + \dfrac {n \pi} 2$
For $n$ even we will then have:
:$\tan \theta = 1$
and:
:$\tan 3 \theta = -1$
For $n$ odd we will then have:
:$\tan \theta = -1$
and:
:$\tan 3 \theta = 1$
It is then directly verified that the Triple Angle Formula for Tangent holds for these special cases where $\tan 2 \theta$ is undefined.
Category:Triple Angle Formula for Tangent
Triple Root is Root of First and Second Derivatives of Equation but not Third Derivative

== Theorem ==
Let $\EE$ be the equation:
:$\map f x = 0$
where $f: \R \to \R$ is a real function.
Let $\xi$ be a multiple root of $\EE$ with a multiplicity of $3$.
Then:
:$\xi$ is a root of the equation $\map {f'} x = 0$
:$\xi$ is a root of the equation $\map {f' '} x = 0$
:$\xi$ is not a root of the equation $\map {f' ' '} x = 0$
where:
:$f': \R \to \R$ is the first derivative of $f$ $x$
:$f' ': \R \to \R$ is the second derivative of $f$ $x$
:$f' ' ': \R \to \R$ is the third derivative of $f$ $x$.
== Proof ==
== Sources ==
* 
* 
Category:Multiple Roots
Category:Differential Calculus
Triple of Triangular Numbers whose Pairwise Sums are Triangular

== Theorem ==
The following triplet of triangular numbers has the property that the sum of each pair of them, and their total, are all triangular numbers:
:$66, 105, 105$
== Proof ==
Throughout we use Closed Form for Triangular Numbers, which gives that the $n$th triangular number can be expressed as:
:$T_n = \dfrac {11 \times 12} 2$
We have:
Then:
== Historical Note ==
== Sources ==
* 
* : $66$
Category:Triangular Numbers
Triple with Product Quadruple the Sum

== Theorem ==
Let $a, b, c \in \N$ such that $a \le b \le c$.
Then the solutions to:
:$a b c = 4 \paren {a + b + c}$
are:
:$\tuple {0, 0, 0}, \tuple {1, 5, 24}, \tuple {1, 6, 14}, \tuple {1, 8, 9}, \tuple {2, 3, 10}, \tuple {2, 4, 6}$
== Proof ==
Suppose $a \ge 4$.
Then:
hence $0 \le a \le 3$.
For $a = 0$, we have $4 \paren {b + c} = 0$.
This forces $b = c = 0$, giving the trivial solution:
:$\tuple {0, 0, 0}$
For $a \ge 1$, note that:
In the second step, is strictly positive.
Hence $a b > 4$ and the final step is justified.
For $a = 1$, we have $b c = 4 \paren {1 + b + c}$.
Since $a b > 4$, $b > 4$.
Suppose $b \ge 9$.
Then:
hence $b \le 8$.
We find the value of $c$ by substituting $a$ and $b$:
:$b = 5: c = \dfrac {4 \paren {1 + 5} } {5 - 4} = 24$
:$b = 6: c = \dfrac {4 \paren {1 + 6} } {6 - 4} = 14$
:$b = 7: c = \dfrac {4 \paren {1 + 7} } {7 - 4} = \dfrac {32} 3$
:$b = 8: c = \dfrac {4 \paren {1 + 8} } {8 - 4} = 9$
and we see that $\tuple {1, 5, 24}, \tuple {1, 6, 14}, \tuple {1, 8, 9}$ are valid solutions.
For $a = 2$, we have $2 b c = 4 \paren {2 + b + c}$.
Since $a b > 4$, $b > 2$.
Suppose $b \ge 5$.
Then:
hence $b \le 4$.
We find the value of $c$ by substituting $a$ and $b$:
:$b = 3: c = \dfrac {4 \paren {2 + 3} } {2 \times 3 - 4} = 10$
:$b = 4: c = \dfrac {4 \paren {2 + 4} } {2 \times 4 - 4} = 6$
and we see that $\tuple {2, 3, 10}, \tuple {2, 4, 6}$ are valid solutions.
For $a = 3$, we have $3 b c = 4 \paren {3 + b + c}$.
Suppose $b \ge 4$.
Then:
hence $b \le 3$.
Since $3 = a \le b$, this forces $b = 3$
We find the value of $c$ by substituting $a$ and $b$:
:$c = \dfrac {4 \paren {3 + 3} } {3 \times 3 - 4} = \dfrac {24} 5$
and we see that it is not a valid solution.
We have considered all possible values of $a$.
Hence the result.