nyuuzyou/wikis huggingface dataset:
Cross-Relation is Equivalence Relation

== Theorem ==
Then $\boxtimes$ is an equivalence relation on $\struct {S_1 \times S_2, \oplus}$.
== Proof ==
=== Reflexivity ===
:$\forall \tuple {x_1, y_1} \in S_1 \times S_2: x_1 \circ y_1 = x_1 \circ y_1 \implies \tuple {x_1, y_1} \boxtimes \tuple {x_1, y_1}$
So $\boxtimes$ is a reflexive relation.
=== Symmetry ===
So $\boxtimes$ is a symmetric relation.
=== Transitivity ===
So $\boxtimes$ is a transitive relation.
All the criteria are therefore seen to hold for $\boxtimes$ to be an equivalence relation.
== Also see ==
=== Examples ===
* Cross-Relation on Natural Numbers is Equivalence Relation
* Cross-Relation on Real Numbers is Equivalence Relation
== Sources ==
* : Chapter $\text {IV}$: Rings and Fields: $\S 20$: The Integers
Category:Cross-Relations
Cross-Relation on Natural Numbers is Equivalence Relation

== Theorem ==
Let $\struct {\N, +}$ be the semigroup of natural numbers under addition.
Let $\struct {\N \times \N, \oplus}$ be the (external) direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$.
The relation $\boxtimes$ defined on $\N \times \N$ by:
:$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$
is an equivalence relation on $\struct {\N \times \N, \oplus}$.
== Proof ==
$\boxtimes$ is an instance of a cross-relation.
We also have that Natural Number Addition is Commutative.
The result therefore follows from Cross-Relation is Equivalence Relation.
== Sources ==
* : $\S 2.5$: Theorem $2.20$
* : Chapter $4$: Mappings: Exercise $9$
Category:Natural Numbers
Category:Examples of Equivalence Relations
Category:Cross-Relations
Cross-Relation on Real Numbers is Equivalence Relation

== Theorem ==
Let $\R^2$ denote the cartesian plane.
Let $\alpha$ denote the relation defined on $\R^2$ by:
:$\tuple {x_1, y_1} \mathrel \alpha \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$
Then $\alpha$ is an equivalence relation on $\R^2$.
=== Geometrical Interpretation ===
== Proof ==
$\alpha$ is an instance of a cross-relation.
We also have that Real Addition is Commutative.
The result therefore follows from Cross-Relation is Equivalence Relation.
== Sources ==
* : Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $4$
Category:Real Numbers
Category:Examples of Equivalence Relations
Category:Cross-Relations
Category:Cross-Relation on Real Numbers is Equivalence Relation
Cross-Relation on Real Numbers is Equivalence Relation/Geometrical Interpretation

== Theorem ==
Let $\R^2$ denote the cartesian plane.
Let $\alpha$ denote the relation defined on $\R^2$ by:
:$\tuple {x_1, y_1} \mathrel \alpha \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$
The equivalence classes of $\alpha$, when interpreted as points in the plane, are the straight lines of slope $1$.
== Proof ==
We have from Cross-Relation on Real Numbers is Equivalence Relation that $\alpha$ is an equivalence relation.
Thus:
Thus each equivalence classes consists of sets of points such that:
:$x - y = c$
That is:
:$y = x + c$
Thus from Equation of Straight Line in Plane, this is the equation of a straight line whose slope is $1$.
== Sources ==
* : Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $4$
Category:Cross-Relation on Real Numbers is Equivalence Relation
Cross-Section of Hyperboloid of Revolution is Circle

== Theorem ==
Let $\HH$ be a hyperboloid of revolution.
Let $\SS$ be a cross-section of $\HH$ taken perpendicular to the axis of rotation of $\HH$.
Then $\SS$ is a circle.
== Proof ==
== Also see ==
* Plane Section of Hyperboloid of Revolution Parallel to Axis of Rotation is Hyperbola
== Sources ==
* 
* 
Category:Hyperboloids of Revolution
Cross Product of Antiparallel Vectors is Zero

== Theorem ==
Let $\mathbf u$ and $\mathbf v$ be antiparallel vectors.
Then:
:$\mathbf u \times \mathbf v = 0$
where $\times$ denotes vector cross product.
== Proof ==
We have that $\mathbf u$ and $\mathbf v$ are antiparallel.
Hence we can define $\mathbf u$ and $\mathbf v$ as:
where $x$, $y$ and $z$ are arbitrary real numbers.
By definition of vector cross product:
:$\mathbf u \times \mathbf v = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ x & y & z \\ -x & -y & -z \end {vmatrix}$
It is seen that the $3$rd row is a multiple of the $2$nd row.
The result then follows from Determinant with Row Multiplied by Constant.
== Sources ==
* 
* 
Category:Antiparallel Vectors
Category:Vector Cross Product
Cross Product of Elements of Standard Ordered Basis

== Theorem ==
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis of Cartesian $3$-space $S$.
Then:
:$\mathbf i \times \mathbf i = \mathbf j \times \mathbf j = \mathbf k \times \mathbf k = 0$
and:
where $\times$ denotes the dot product.
== Proof ==
From Cross Product of Vector with Itself is Zero:
:$\mathbf i \times \mathbf i = \mathbf j \times \mathbf j = \mathbf k \times \mathbf k = 0$
Then we can take the definition of cross product:
:$\mathbf a \times \mathbf b = \begin {vmatrix}
\mathbf i & \mathbf j & \mathbf k \\
a_i & a_j & a_k \\
b_i & b_j & b_k \\
\end {vmatrix} = \mathbf a \times \mathbf b = \paren {a_j b_k - a_k b_j} \mathbf i - \paren {a_i b_k - a_k b_i} \mathbf j + \paren {a_i b_j - a_j b_i} \mathbf k$
and note that:
Hence:
The remaining identities follow from Vector Cross Product is Anticommutative:
== Sources ==
* : Chapter $\text {II}$: The Products of Vectors: $4$. The Vector Product: $(2.16)$, $(2.17)$
* : Chapter $\text I$: Vector Algebra: $\S 3$: $(7)$, $(8)$
Category:Vector Cross Product
Category:Standard Ordered Bases
Cross Product of Parallel Vectors

== Theorem ==
Let $\mathbf a$ and $\mathbf b$ be vector quantities which are parallel.
Let $\mathbf a \times \mathbf b$ denote the cross product of $\mathbf a$ with $\mathbf b$.
Then:
:$\mathbf a \times \mathbf b = \mathbf 0$
where $\mathbf 0$ denotes the zero vector.
== Proof ==
By definition of cross product:
:$\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the length of $\mathbf a$
:$\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction
:$\hat {\mathbf n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right-hand rule.
By definition of parallel, the direction of $\mathbf a$ is the same as the direction of $\mathbf b$
Hence when $\mathbf a$ and $\mathbf b$ are parallel, $\theta = 0$ by definition.
Hence from Sine of Zero is Zero, $\sin \theta = 0$.
Hence the result.
== Sources ==
* : Chapter $\text {II}$: The Products of Vectors: $4$. The Vector Product: $(2.14)$
* : Chapter $\text I$: Vector Algebra: $\S 3$
Category:Vector Cross Product
Cross Product of Perpendicular Vectors

== Theorem ==
Let $\mathbf a$ and $\mathbf b$ be vector quantities which are perpendicular.
Let $\mathbf a \times \mathbf b$ denote the cross product of $\mathbf a$ with $\mathbf b$.
Then:
:$\mathbf a \times \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the length of $\mathbf a$
:$\hat {\mathbf n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right-hand rule.
== Proof ==
By definition of cross product:
:$\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the length of $\mathbf a$
:$\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction
:$\hat {\mathbf n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right-hand rule.
When $\mathbf a$ and $\mathbf b$ are perpendicular, $\theta = 90 \degrees$ by definition.
Hence from Sine of Right Angle, $\sin \theta = 1$.
Hence the result.
== Sources ==
* : Chapter $\text {II}$: The Products of Vectors: $4$. The Vector Product: $(2.15)$
Category:Vector Cross Product
Cross Product of Vector with Itself is Zero

== Theorem ==
Let $\mathbf x$ be a vector in a vector space of $3$ dimensions:
:$\mathbf x = x_i \mathbf i + x_j \mathbf j + x_k \mathbf k$
Then:
:$\mathbf x \times \mathbf x = \mathbf 0$
where $\times$ denotes vector cross product.
== Proof 1 ==
== Proof 2 ==
Category:Cross Product of Vector with Itself is Zero
Category:Vector Cross Product
Crossbar Theorem

== Theorem ==
Let $\triangle ABC$ be a triangle.
Let $D$ be a point in the interior of $\triangle ABC$.
Then there exists a point $E$ such that $E$ lies on both $AD$ and $BC$.
== Proof ==
:
 $BC$ does not meet ray $\overrightarrow {AD}$.
Either $BC$ meets line $\overleftrightarrow {AD}$ or it does not.
If it meets $\overleftrightarrow {AD}$, by the Line Separation Property it must meet the ray opposite to $\overrightarrow {AD}$ at a point $E \ne A$.
According to Proposition 3.8 (b), $E$ is not in the interior of $\angle CAB$.
Point $B$ does not lie on $\overleftrightarrow {AD}$; this is because $D \in \operatorname {int} \angle CAB$, so $D$ and $C$ lie on the same side of $\overleftrightarrow {AB}$, so $D \notin \set {\overleftrightarrow {AB} }$, so $B \notin \set {\overleftrightarrow {AB} }$.
Thus, since $E \in \set {\overleftrightarrow {AD} }$, we have:
:$E \ne B$
By the same reasoning with $C$ and $B$ interchanged, we have:
:$E \ne C$
Since $E \in BC$ and $E$ is not an endpoint, we have:
:$B ∗ E ∗ C$
Thus by Proposition 3.7, $E$ is in the interior of $\angle CAB$, which is a contradiction.
Thus $\overleftrightarrow {AD}$ does not meet $BC$ at all; that is, $B$ and $C$ are on the same side of $\overleftrightarrow {AD}$.
By B-2, we have a point $E$ such that:
:$C ∗ A ∗ E$
By Lemma 3.2.2, $C$ and $E$ are on opposite sides of $\overleftrightarrow {AD}$.
Thus, by B-4(iii), E and B are on opposite sides of $\overleftrightarrow {AD}$.
But by Proposition 3.8(c), $B$ is on the interior of $\angle DAE$, so $E$ and $B$ are on the same side of $\overleftrightarrow {AD}$
This is a contradiction.
Thus $\overrightarrow {AD}$ meets $BC$.
Category:Triangles
Category:Named Theorems
Cube Function is Odd

== Theorem ==
The cube function on the real numbers:
:$\forall x \in \R: \map f x = x^3$
is an odd function.
== Proof ==
Hence the result by definition of odd function.
== Sources ==
* : Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(h)}$ Even and Odd Functions
Category:Examples of Odd Functions
Cube Modulo 9

== Theorem ==
Let $x \in \Z$ be an integer.
Then one of the following holds:
== Proof ==
Let $x$ be an integer.
There are three cases to consider:
:$(1): \quad x \equiv 0 \pmod 3$: we have $x = 3 k$
:$(2): \quad x \equiv 1 \pmod 3$: we have $x = 3 k + 1$
:$(3): \quad x \equiv 2 \pmod 3$: we have $x = 3 k + 2$
Then:
== Sources ==
* : Chapter $2$: Divisibility Theory in the Integers: $2.1$ The Division Algorithm: Problems $2.1$: $3 \ \text{(c)}$
Category:Modulo Arithmetic
Category:Cube Numbers
Cube Number/Sequence

== Definition ==
The sequence of positive cube numbers begins:
:$1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, \ldots$
== Sources ==
* : Tables: $1$ The First $100$ Triangular Numbers, Squares and Cubes
* : Tables: $1$ The First $100$ Triangular Numbers, Squares and Cubes
Category:Cube Numbers
Cube Number as Difference between Squares of Triangular Numbers

== Theorem ==
Let $n \in \Z_{>0}$ be a positive integer.
Then:
:$n^3 = {T_n}^2 - {T_{n - 1} }^2$
where $T_n$ denotes the $n$th triangular number.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Problems $1.1$: $4$
* : $15$
* : $15$
Category:Cube Number as Difference between Squares of Triangular Numbers
Category:Triangular Numbers
Category:Cube Numbers
Cube Number as Sum of Three Consecutive Odd Squares

== Theorem ==
:$1331 = 11^3 = 19^2 + 21^2 + 23^2$
No other such sequence of $3$ consecutive odd squares has the same property.
== Proof ==
Any sequence of $3$ consecutive odd integers that have squares that sum to a cube would satisfy:
:$m^3 = \paren {n - 2}^2 + n^2 + \paren {n + 2}^2$
where $n$ is the middle number of the sequence, with $m, n \in \Z$.
Expanding the :
Substituting $x = 3 m$ and $y = 9 n$:
which is an elliptic curve.
According to LMFDB, this elliptic curve has exactly $5$ lattice points:
:$\tuple {6, 0}, \tuple {10, \pm 28}, \tuple {33, \pm 189}$
which correspond to these values of $n$:
:$0, \pm \dfrac {28} 9, \pm 21$
$\pm \dfrac {28} 9$ are not integers.
The other possibilities are:
:the trivial $0 = \paren {-2}^2 + 0^2 + 2^2$
:the less trivial $11^3 = \paren {-23}^2 + \paren {-21}^2 + \paren {-19}^2$
The first is not a set of consecutive odd squares.
The second gives the same set of consecutive odd squares.
Hence there are no more solutions.
== Historical Note ==
== Sources ==
* : $1331$
* 
Category:Cube Numbers
Category:Sums of Squares
Category:1331
Cube Number multiplied by Cube Number is Cube

== Theorem ==
Let $a, b \in \N$ be natural numbers.
Let $a$ and $b$ be cube numbers.
Then $a b$ is also a cube number.
== Proof ==
By the definition of cube number:
:$\exists r \in \N: r^3 = a$
:$\exists s \in \N: s^3 = b$
Thus:
Thus:
:$\exists k = r s \in \N: a = k^3$
Hence the result by definition of cube number.
== Sources ==
* : Book $\text{IX}$. Propositions
Category:Euclidean Number Theory
Cube Root/Examples/2

== Decimal Expansion ==
The decimal expansion of the cube root of $2$ starts:
:$\sqrt [3] 2 \approx 1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$
== Sources ==
* : Table $1.1$. Mathematical Constants
* : $\S 1$: Special Constants: $1.6$
* : $1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$
* : $1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$
Category:Examples of Cube Roots
Category:2
Cube Root/Examples/3

== Decimal Expansion ==
The decimal expansion of the cube root of $3$ starts:
:$\sqrt [3] 3 \approx 1 \cdotp 44224 \, 95703 \, 07408 \, 3823 \ldots$
== Sources ==
* : Table $1.1$. Mathematical Constants
* : $\S 1$: Special Constants: $1.7$
Category:Examples of Cube Roots
Category:3
Cube Root of Unity if Modulus is 1 and Real Part is Minus Half

== Theorem ==
Let $z \in \C$ be a complex number such that:
:$\cmod z = 1$
:$\Re \paren z = -\dfrac 1 2$
where:
:$\cmod z$ denotes the complex modulus of $z$
:$\Re \paren z$ denotes the real part of $z$.
Then:
:$z^3 = 1$
== Proof ==
Let $z = x + i y$.
From $\Re \paren z = -\dfrac 1 2$:
:$x = -\dfrac 1 2$
by definition of the real part of $z$.
Then:
Thus:
:$z = -\dfrac 1 2 \pm \dfrac {\sqrt 3} 2$
and the result follows from Cube Roots of Unity.
== Sources ==
* : $\S 3$. Roots of Unity: Exercise $4$.
Category:Cube Roots of Unity
Cube as Sum of Sequence of Centered Hexagonal Numbers

== Theorem ==
:$C_n = \ds \sum_{i \mathop = 1}^n H_i$
where:
:$C_n$ denotes the $n$th cube number
:$H_i$ denotes the $i$th centered hexagonal number.
== Proof ==
From Closed Form for Centered Hexagonal Numbers:
:$H_n = 3 n \paren {n - 1} + 1$
Hence:
== Examples ==
== Sources ==
* : $64$
* : $64$
Category:Centered Hexagonal Numbers
Category:Cube Numbers
Category:Cube as Sum of Sequence of Centered Hexagonal Numbers
Cube of 11 is Palindromic

== Theorem ==
::$11^3 = 1331$
== Proof ==
From Square of 11 is Palindromic:
:$11^2 = 121$
Thus:
 121
x 11
-----
 121
1210
-----
1331
== Sources ==
* : $121$
* : $121$
Category:Cube Numbers
Category:11
Category:1331
Category:Palindromic Numbers
Cube of 180 is Sum of Sequence of Consecutive Cubes

== Theorem ==
:$180^3 = \ds \sum_{k \mathop = 6}^{69} k^3$
That is:
:$180^3 = 6^3 + 7^3 + \cdots + 67^3 + 68^3 + 69^3$
== Proof ==
== Sources ==
* : $180$
* : $180$
Category:Cube Numbers
Category:180
Cube of 20 is Sum of Sequence of 4 Consecutive Cubes

== Theorem ==
:$20^3 = \ds \sum_{k \mathop = 11}^{14} k^3$
That is:
:$20^3 = 11^3 + 12^3 + 13^3 + 14^3$
== Proof ==
== Sources ==
* : $8000$
* : $8000$
Category:Cube Numbers
Category:20
Category:8000
Cube of 3334

== Interesting Property ==
The cube of $3334$ has the following interesting property:
:$3334^3 = 037 \, 059 \, 263 \, 704$
while:
:$0370 + 5926 + 3704 = 10 \, 000$
== Historical Note ==
== Sources ==
* : $3334$
* : $3334$
Category:Cube Numbers
Category:3334
Cube of 71 is Odd Integers in Sequence

== Theorem ==
The cube of $71$, when expressed in decimal notation, is the odd integers from $3$ to $11$ written in sequence:
::$71^3 = 357 \, 911$
== Proof ==
Verified by direct calculation.
== Historical Note ==
== Sources ==
* : $71$
* : $71$
Category:Cube Numbers
Category:Odd Integers
Category:71
Cube of 71 is Odd Integers in Sequence/Historical Note

== Historical Note on Cube of 71 is Odd Integers in Sequence ==
The interesting phenomenon that $71^3$ consists of the odd integers from $3$ to $11$ in sequence is reported by in as being noted by a contributor to called James Davies, writing in Volume $13$.
An online listing of the articles for that journal has revealed an article titled An Evening With Pi by , but further investigation will need to be done to establish the exact nature of this.
== Sources ==
* : $71$
* : $71$
Category:Historical Notes
Cube of Complex Number

== Theorem ==
Let $z = a + i b$ be a complex number.
Then its cube is given by:
:$z^3 = a^3 - 3 a b^2 + i \paren {3 a^2 b - b^3}$
== Proof ==
== Sources ==
* : $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Powers: $3.7.19$
Category:Complex Numbers
Category:Cube Function
Cube of Prime has 4 Positive Divisors

== Theorem ==
Let $n \in \Z_{>0}$ be a positive integer which is the cube of a prime number.
Then $n$ has exactly $4$ positive divisors.
== Proof ==
Let $n = p^3$ where $p$ is prime.
The positive divisors of $n$ are:
:$1, p, p^2, p^3$
This result follows from Divisors of Power of Prime.
== Sources ==
* : $33$
Category:Cube Numbers
Cube of Prime is Multiplicatively Perfect

== Theorem ==
Let $n \in \Z_{>0}$ be a positive integer which is the cube of a prime number.
Then $n$ is multiplicatively perfect.
== Proof ==
Let $n = p^3$ where $p$ is prime.
From Cube of Prime has 4 Positive Divisors, the positive divisors of $n$ are:
:$1, p, p^2, p^3$
Thus the product of all the divisors of $n$ is:
:$1 \times p \times p^2 \times p^3 = p^6 = n^2$
Hence the result, by definition of multiplicatively perfect.
== Sources ==
* : $33$
Category:Cube Numbers
Category:Multiplicatively Perfect Numbers
Cube of n minus 23 Greater than Square of (4n-7)

== Theorem ==
Let $n \in \Z$ such that $n \ge 12$.
Then:
:$n^3 - 23 > \paren {4 n - 7}^2$
== Proof ==
The proof proceeds by induction.
For all $n \in \Z$ such that $n \ge 12$, let $\map P n$ be the proposition:
:$n^3 - 23 > \paren {4 n - 7}^2$
=== Basis for the Induction ===
$\map P {12}$ is the case:
Thus $\map P {12}$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that if $\map P k$ is true, where $k \ge 12$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
:$k^3 - 23 > \paren {4 k - 7}^2$
from which it is to be shown that:
:$\paren {k + 1}^3 - 23 > \paren {4 \paren {k + 1} - 7}^2$
=== Induction Step ===
This is the induction step:
Then we calculate:
Thus we have:
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
:$\forall n \in \Z_{\ge 12}: n^3 - 23 > \paren {4 n - 7}^2$
== Sources ==
* 
* 
Category:Inequalities
Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes

== Theorem ==
:$351 \, 120^3$ can be represented as the sum of $3$, $4$, $5$, $6$, $7$ or $8$ cubes.
== Proof ==
These representations are not necessarily unique.
== Additional Results ==
We also have:
So $351120^3$ can also be expressed as a sum of $9$ or $10$ cubes.
These equations all stem from:
showing that $351 \, 120$ is not the smallest number with this property.
Moreover, using $3^3 + 4^3 + 5^3 = 6^3$, this result could still be further extended:
which is a sum of $11$ to $19$ cubes.
Using $1^3 + 6^3 + 8^3 = 9^3$ and $\paren {2 n}^3 = 8 \times n^3$, we can express $12^3$ as more and more cubes.
== Historical Note ==
== Sources ==
* : $351,120$
* : $351,120$
Category:Cube Numbers
Category:351,120
Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes/Historical Note

== Historical Note on Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes ==
This result can be found in by from $1986$, but it is a mystery as to why it was presented.
There appears to be absolutely nothing special about $351 \, 120$, as there are many cubes far smaller which possess the same property.
It is possible that misinterpreted something that caught his eye, but googling for properties of $351 \, 120$ reveals that it is spectacularly uninteresting.
Category:Historical Notes
Cube which is One Less than a Square

== Theorem ==
$8$ is the only cube number which is $1$ less than a square:
:$2^3 + 1 = 3^2$
== Proof ==
This is a specific instance of 1 plus Perfect Power is not Prime Power except for 9.
== Sources ==
* : $8$
* : $8$
Category:8
Category:9
Category:Square Numbers
Category:Cube Numbers
Cubes which are Sum of Three Cubes

== Theorem ==
The following cube numbers can be expressed as the sum of $3$ positive cube numbers:
:$6^3, 9^3, 12^3, 18^3, 19^3, 20^3, 24^3, 25^3, \ldots$
The associated cube roots:
:$6, 9, 12, 18, 19, 20, 24, 25, \ldots$
== Examples ==
== Sources ==
* 
* : $216$
* : $729$
* : $216$
* : $729$
Category:Sums of Cubes
Cubic Fibonacci Numbers

== Theorem ==
The only Fibonacci numbers which are also cubes are:
:$F_1 = 1 = 1^3$
:$F_6 = 8 = 2^3 = 3 + 5$
== Proof ==
== Sources ==
* : $8$
* : $8$
Category:8
Category:Cube Numbers
Category:Fibonacci Numbers
Cubical Parabola has Point of Inflection

== Theorem ==
The cubical parabola has a point of inflection.
When the equation for the cubical parabola is in the form $y = a x^3$, this point of inflection is the origin.
== Proof ==
By definition, there is a point of inflection at a point where $x = \xi$ the first derivative has either a local maximum or a local minimum at $\xi$.
We note that $3 a x^2 \ge 0$ is the equation for the parabola.
From Square of Non-Zero Real Number is Strictly Positive, we note that:
:$3 a x^2 = 0 \iff x = 0$
and:
:$\forall x \in \R: x \ne 0 \implies x^2 > 0$
So:
:if $a > 0$ then $3 a x^2$ has a local minimum at $x = 0$
:if $a < 0$ then $3 a x^2$ has a local maximum at $x = 0$.
Hence the result.
== Sources ==
* 
* 
Category:Cubical Parabola
Category:Points of Inflection
Cuboid with Integer Edges and Face Diagonals

== Theorem ==
The smallest cuboid whose edges and the diagonals of whose faces are all integers has edge lengths $44$, $117$ and $240$.
Its space diagonal, however, is not an integer.
== Proof ==
The edges are given as having lengths $44$, $117$ and $240$.
The faces are therefore:
:$44 \times 117$
:$44 \times 240$
:$117 \times 240$
The diagonals of these faces are given by Pythagoras's Theorem as follows:
However, its space diagonal is calculated as:
which, as can be seen, is not an integer.
== Historical Note ==
== Sources ==
* : $44$
* : $44$
Category:Cuboids
Cumulative Distribution Function/Examples/Arbitrary Example 1

== Example of Cumulative Distribution Function ==
Consider a population consisting of children the state of whose teeth is being monitored.
The following table consists of a count of the number of teeth with dental caries in a group of $50$ schoolchildren:
:$\begin {array} {|l|l|}
\hline
\text {Number of Teeth} & \text {Number of Children} \\
\hline
0 & 27 \\
1 & 12 \\
2 & 6 \\
3 & 4 \\
6 & 1 \\
\hline
\end {array}$
The values of the cumulative distribution function:
== Sources ==
* 
* 
Category:Examples of Cumulative Distribution Functions