nyuuzyou/wikis huggingface dataset:
Archimedes' Limits to Value of Pi/Lemma 4

== Lemma for Archimedes' Limits to Value of Pi ==
:$\pi > \dfrac {223} {71}$
== Proof ==
;Lower bound
:
Let $\triangle BCA$ be a right triangle inscribed in the same circle $O$ from above with diameter $AB = 1$.
We define the following angles:
Using the same argument as we used previously, $\triangle BCA$ has sides in the ratio:
:$BC : AB : AC = 1 : 2 : \sqrt 3$
$BC$ subtends a central angle of $2 \theta_1 = 120 \degrees$, so it corresponds to one side of an inscribed hexagon.
The total perimeter $p$ of the inscribed hexagon is:
:$p = 6 \cdot BC$
The ratio of the perimeter of the inscribed hexagon to the diameter of the circle is:
:$\dfrac p {AB} = \dfrac {6 BC} {AB} = 6 \cdot \dfrac {BC} {AB}$
Since $AB = 1$, the perimeter of the inscribed hexagon is:
:$p = 6 \cdot \dfrac 1 2 = 3$
This is an initial lower bound on $\pi$.
We now continue by doubling the number of sides of the inscribed regular polygon to refine our estimate.
We have:
We will now use the following rational approximation for $\sqrt 3$, whose decimal value is approximately $1.73205$:
:$\dfrac {AC} {BC} = \cot \theta_1 < \dfrac {1351} {780}$
This approximation is about $1.7320513$, which is slightly larger than the true value.
The cosecant of $\theta_1$ is $2$ so:
:$\dfrac {AB} {BC} = \dfrac {1560} {780}$
Thus, the initial lower bound estimate for $\pi$ is:
;Second Iteration Lower Bound Calculation
We see that:
:$3013 \tfrac 3 4 = 3013.75$ is higher than the square root of $9 \, 082 \, 321$ which is approximately $3013.689$
Thus, our second lower bound estimate for $\pi$ is:
;Third Iteration Lower Bound Calculation
We see that:
:$1838 \tfrac 9 {11} = 1838.818$ is higher than the square root of $3 \, 380 \, 929$ which is approximately $1838.730$
Thus, our third lower bound estimate for $\pi$ is:
;Fourth Iteration Lower Bound Calculation
We see that:
:$1009 \tfrac 1 6 > 1009.166$ is higher than the square root of $1 \, 018 \, 405$ which is approximately $1009.161$.
Thus, our fourth lower bound estimate for $\pi$ is:
;Fifth Iteration Lower Bound Calculation
We see that:
:$2017 \tfrac 1 4 = 2017.25$ is higher than the square root of $4 \, 069 \, 284.03$ which is approximately $2017.247$.
Thus, our fifth lower bound estimate for $\pi$ is:
Category:Archimedes' Limits to Value of Pi
Archimedes' Limits to Value of Pi/Trigonometric Proof

== Theorem ==
== Proof ==
Let $O$ be a circle with diameter $AB = 1$
Let $Q$ be the circumference of $O$.
By the definition of $\pi$:
:$\dfrac Q {AB} = \pi$
Since $AB = 1$ then $Q$ is exactly $\pi$.
Using regular polygons inscribed and circumscribed about the circle $O$, we aim to demonstrate that the bounds on $\pi$ are:
:$\dfrac {223} {71} < \pi < \dfrac {22} {7}$
=== Lemma 1 ===
=== Lemma 2 ===
=== Lemma 3 ===
=== Lemma 4 ===
Combining the two results from the fifth iteration of and , we have:
Expressing this in decimal:
:$3.1408 < \pi < 3.1429$
== Historical Note ==
Category:Archimedes' Limits to Value of Pi
Archimedes' Principle

== Physical Law ==
Let $V$ be a compact body with a piecewise smooth boundary, immersed in an incompressible fluid $F$.
Then the net buoyancy $\mathbf F$ effected upon $V$ by $F$ is equal to the weight of $F$ displaced.
The line of action of $\mathbf F$ passes through the center of gravity of the displaced volume of $F$.
This is often quoted (and probably better considered) as the informal statement:
:A body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid.
This is known as Archimedes' principle.
== Proof ==
Let $V$ be the immersed body.
Let $S = \partial V$ be its boundary.
Recall Gauss's Theorem for a smooth vector field $\mathbf F$ defined over $V$:
:$(1): \quad \ds \oint_S \mathbf F \cdot \rd \mathbf S = \int_V \nabla \cdot \mathbf F \rd \mathbf V$
provided that $\partial V$ is piecewise smooth and compact.
The pressure on $S$ depends only on the depth of $V$ within $F$.
Accounting for atmospheric pressure $p_0$, $\mathbf F$ is given by:
:$p = -\rho g z + p_0$
where:
:$\rho$ is the density of the fluid
:$g = 9.81 \ldots$ is the gravitational acceleration
:$z$ is the vertical displacement.
Letting $\mathbf F = -p \cdot \mathbf k$ (with $\mathbf k$ a unit vector in the $z$ direction) we see that the of $(1)$ becomes the buoyancy force acting on the object, for it is the sum over the surface of the $z$ component of the pressure.
Clearly $\nabla \cdot \mathbf F = \rho g$, so we have:
:$\ds \int_V \nabla \cdot \mathbf F \rd V = \rho g \int_V \rd V = \rho g V$
where we have let $V$ denote the scalar volume of $V$.
Note that we have assumed incompressibility and thus constant density of the fluid.
This is precisely the weight of the fluid in the volume $V$ Ref?.
The proof is complete.
== Also known as ==
== Also see ==
Not to be confused with the Archimedean Principle.
== Historical Note ==
== Sources ==
* : Chapter $\text {A}.5$: Archimedes (ca. B.C.)
* 
* 
* 
* 
Category:Archimedes' Principle
Category:Buoyancy
Category:Hydrostatics
Category:Fluid Mechanics
Arcsecant Logarithmic Formulation

== Theorem ==
Let $x$ be a real number. 
Let $x \in \hointl \gets {-1} \cup \hointr 1 \to$.
Then: 
:$\ds \arcsec x = -i \map \Ln {i \sqrt {1 - \frac 1 {x^2} } + \frac 1 x}$
where:
:$\arcsec$ is the arcsecant function
:$\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\hointl {-\pi} \pi$. 
== Proof ==
== Also see ==
* Arcsine Logarithmic Formulation
* Arccosine Logarithmic Formulation
* Arctangent Logarithmic Formulation
* Arccotangent Logarithmic Formulation
* Arccosecant Logarithmic Formulation
Category:Arcsecant Function
Arcsecant of Reciprocal equals Arccosine

== Theorem ==
Everywhere that the function is defined:
:$\map \arcsec {\dfrac 1 x} = \arccos x$
where $\arccos$ and $\arcsec$ denote arccosine and arcsecant respectively.
== Proof ==
Category:Arccosine Function
Category:Arcsecant Function
Category:Reciprocals
Arcsine Function in terms of Gaussian Hypergeometric Function

== Theorem ==
:$\arcsin x = x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2}$
where:
:$x$ is a real number with $\size x \le 1$
:$\arcsin$ denotes the arcsine function
:$\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {x^k} {k!}$
== Proof ==
== Also presented as ==
This result can also be seen presented in the form:
:$\map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2} = \dfrac {\arcsin x} x$
== Sources ==
* : $\S 31$: Hypergeometric Functions: Special Cases: $31.8$
Category:Arcsine Function
Category:Gaussian Hypergeometric Function
Category:Hypergeometric Functions
Arcsine Logarithmic Formulation

== Theorem ==
For any real number $x: -1 \le x \le 1$:
:$\arcsin x = \dfrac 1 i \map \ln {i x + \sqrt {1 - x^2} }$
where $\arcsin x$ is the arcsine and $i^2 = -1$.
== Proof ==
Assume $y \in \R$ where $-\dfrac \pi 2 \le y \le \dfrac \pi 2$.
== Also see ==
* Arccosine Logarithmic Formulation
* Arctangent Logarithmic Formulation
* Arccotangent Logarithmic Formulation
* Arcsecant Logarithmic Formulation
* Arccosecant Logarithmic Formulation
Category:Arcsine Function
Arcsine as Integral

== Theorem ==
:$\ds \map \arcsin x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$
== Proof ==
=== Lemma 1 ===
=== Lemma 2 ===
:$\ds \map \arcsin x = \map {\arcsin_A} x = \map {\arcsin_G} x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$
== Also see ==
* Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form
Category:Arcsine Function
Category:Primitive of Reciprocal of Root of a squared minus x squared
Category:Arcsine as Integral
Arcsine as Integral/Lemma 1

== Lemma ==
Let $\sin_A$ be the analytic sine function for real numbers.
Let $\arcsin_A$ denote the real arcsine function.
Then:
:$\ds \map {\arcsin_A} x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$
== Proof ==
For this proof only, let $\sin_A$ be the analytic sine function.
Consider:
:$\ds \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$
Let:
:$x = \sin_A \theta \iff \theta = \map {\arcsin_A} x$
Then:
Category:Arcsine as Integral
Arcsine as Integral/Lemma 2

== Lemma ==
Let $\sin_G$ be the geometric sine.
$\arcsin_G$ is the inverse of this function.
:$\ds \map {\arcsin_G} x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$
== Proof ==
This result will be used in proving Derivative of Sine Function in the geometric case.
So we can not use the same reasoning as Lemma 1 because our logic would be circular.
:
Let $\theta$ be the length of the arc associated with the angle on the circle of radius $1$.
By definition of arcsine:
:$y = \sin \theta \iff \theta = \arcsin y$
We have that arc length is always positive. 
For negative $y$, the $\arcsin$ function is defined as being the negative of the arc length.
This makes the $\arcsin$ function and the $\sin$ function odd, and puts us in line with mathematical convention:
:Inverse Sine is Odd Function.
:Sine Function is Odd
Without this convention, the derivative of the $\sin$ function would not be continuous. 
Now:
Then:
Category:Arcsine as Integral
Arcsine in terms of Arctangent

== Theorem ==
:$\arcsin x = \map \arctan {\dfrac x {\sqrt {1 - x^2} } }$
where $x$ is a real number with $-1 < x < 1$.
== Proof ==
Let: 
:$\theta = \arcsin x$
Then by the definition of arcsine:
:$x = \sin \theta$
and: 
:$-\dfrac \pi 2 < \theta < \dfrac \pi 2$
Then: 
== Also see ==
* Arccosine in terms of Arctangent
* Arctangent in terms of Arcsine
Category:Arcsine Function
Category:Arctangent Function
Arcsine in terms of Twice Arctangent

== Theorem ==
:$\arcsin x = 2 \map \arctan {\dfrac x {1 + \sqrt {1 - x^2} } }$
where $x$ is a real number with $-1 < x < 1$.
== Proof ==
Let: 
:$\theta = \arcsin x$
Then by the definition of arcsine:
:$x = \sin \theta$
and: 
:$-\dfrac \pi 2 < \theta < \dfrac \pi 2$
Then:
Category:Arcsine Function
Category:Arctangent Function
Arcsine of One is Half Pi

== Theorem ==
:$\arcsin 1 = \dfrac \pi 2$
where $\arcsin$ is the arcsine function. 
== Proof ==
By definition, $\arcsin$ is the inverse of the restriction of the sine function to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Therefore, if: 
:$\sin x = 1$
and $-\dfrac \pi 2 \le x \le \dfrac \pi 2$, then $\arcsin 1 = x$.
From Sine of Right Angle, we have that: 
:$\sin \dfrac \pi 2 = 1$
We therefore have: 
:$\arcsin 1 = \dfrac \pi 2$ 
Category:Arcsine Function
Arcsine of Reciprocal equals Arccosecant

== Theorem ==
Everywhere that the function is defined:
:$\map \arcsin {\dfrac 1 x} = \arccsc x$
where $\arcsin$ and $\arccsc$ denote arcsine and arccosecant respectively.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.77$
Category:Arcsine Function
Category:Arccosecant Function
Category:Reciprocals
Arcsine of Zero is Zero

== Theorem ==
:$\arcsin 0 = 0$
where $\arcsin$ is the arcsine function. 
== Proof ==
By definition, $\arcsin$ is the inverse of the restriction of the sine function to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Therefore, if: 
:$\sin x = 0$
and $-\dfrac \pi 2 \le x \le \dfrac \pi 2$, then $\arcsin 0 = x$. 
From Sine of Zero is Zero, we have that: 
:$\sin 0 = 0$
We have $-\dfrac \pi 2 < 0 < \dfrac \pi 2$, so: 
:$\arcsin 0 = 0$
Category:Arcsine Function
Arctangent Function in terms of Gaussian Hypergeometric Function

== Theorem ==
:$\arctan x = x \map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2}$
where:
:$x$ is a real number with $\size x < 1$
:$\arctan$ denotes the arctangent function
:$\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {x^k} {k!}$
== Proof ==
== Also presented as ==
This result can also be seen presented in the form:
:$\map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2} = \dfrac {\arctan x} x$
== Sources ==
* : $\S 31$: Hypergeometric Functions: Special Cases: $31.9$
Category:Arctangent Function
Category:Gaussian Hypergeometric Function
Category:Hypergeometric Functions
Arctangent Logarithmic Formulation

== Theorem ==
For any real number $x$:
:$\arctan x = \dfrac 1 2 i \map \ln {\dfrac {1 - i x} {1 + i x} }$
where $\arctan x$ is the arctangent and $i^2 = -1$.
== Proof ==
Assume $y \in \R$, $ -\dfrac \pi 2 \le y \le \dfrac \pi 2 $.
== Also see ==
* Arcsine Logarithmic Formulation
* Arccosine Logarithmic Formulation
* Arccotangent Logarithmic Formulation
* Arcsecant Logarithmic Formulation
* Arccosecant Logarithmic Formulation
Category:Arctangent Function
Arctangent in terms of Arcsine

== Theorem ==
:$\arctan x = \map \arcsin {\dfrac x {\sqrt {1 + x^2} } }$
where $x$ is a real number.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Arcsine in terms of Arctangent
* Arccosine in terms of Arctangent
Category:Arctangent in terms of Arcsine
Category:Arcsine Function
Category:Arctangent Function
Arctangent in terms of Arcsine/Proof 2

== Theorem ==
== Proof ==
From Pfaff's Transformation:
:$\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$
where $\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$.
We have:
Therefore:
:$\map \arctan x = \map \arcsin {\dfrac x {\sqrt {1 + x^2} } }$
== Sources ==
* : Chapter $\text {2}$. The Hypergeometric Functions
Category:Arctangent in terms of Arcsine
Category:Gaussian Hypergeometric Function
Category:Pfaff's Transformation‎
Arctangent is of Exponential Order Zero

== Theorem ==
Let $\arctan: \R \to \openint {-\dfrac \pi 2} {\dfrac \pi 2}$ be the real arctangent.
Then $\arctan$ is of exponential order $0$.
== Proof ==
Follows from Function with Limit at Infinity of Exponential Order Zero.
Category:Exponential Order
Arctangent of Imaginary Number

== Theorem ==
Let $x$ belong to the open real interval $\openint {-1} 1$.
Then:
:$\map {\tan^{-1} } {i x} = \dfrac i 2 \map \ln {\dfrac {1 + x} {1 - x} }$
where $\tan$ is the complex tangent function, $\ln$ is the real natural logarithm, and $i$ is the imaginary unit.
== Proof ==
Let $y = \map {\tan^{-1} } {i x}$.
Let $x = \tanh \theta$, then $\theta = \tanh^{-1} x$.
Category:Complex Numbers
Category:Tangent Function
Arctangent of One

== Theorem ==
:$\map \arctan 1 = \dfrac \pi 4$
== Proof ==
By definition, $\arctan$ is the inverse of the tangent function's restriction to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$. 
By Tangent of $45 \degrees$: 
:$\tan \dfrac \pi 4 = 1$.
As $\dfrac \pi 4 \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$, we have by the definition of an inverse function: 
:$\map \arctan 1 = \dfrac \pi 4$
Category:Arctangent Function
Arctangent of Reciprocal equals Arccotangent

== Theorem ==
Everywhere that the function is defined:
:$\map \arctan {\dfrac 1 x} = \arccot x$
where $\arctan$ and $\arccot$ denote arctangent and arccotangent respectively.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.79$
Category:Arctangent Function
Category:Arccotangent Function
Category:Reciprocals
Arctangent of Root 3 over 3

== Theorem ==
:$\map \arctan {\dfrac {\sqrt 3} 3} = \dfrac \pi 6$
== Proof ==
By definition, $\arctan$ is the inverse of the tangent function's restriction to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$. 
By Tangent of $30 \degrees$: 
:$\tan \dfrac \pi 6 = \dfrac {\sqrt 3} 3$.
As $\dfrac \pi 6 \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$, we have by the definition of an inverse function: 
:$\map \arctan {\dfrac {\sqrt 3} 3} = \dfrac \pi 6$
Category:Arctangent Function
Arctangent of Zero is Zero

== Theorem == 
:$\arctan 0 = 0$
== Proof ==
By definition, $\arctan$ is the inverse of the tangent function's restriction to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$. 
By Tangent of Zero:
:$\tan 0 = 0$
As $0 \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$, we have $\arctan 0 = 0$ by the definition of an inverse function.
Category:Arctangent Function
Are All Perfect Numbers Even?

== Open Question ==
By the Theorem of Even Perfect Numbers, it is known that an even number is perfect it is of the form:
:$2^{n - 1} \paren {2^n - 1}$
where $2^n - 1$ is prime.
It is not known whether there exist any odd perfect numbers.
None have ever been found.
== Progress ==
== Sources ==
* : Preface
* : $\text {3-5}$ The Use of Computers in Number Theory: Conjecture $2$
* : $2$
* : $28$
* 
* : Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes
* 
* : $2$
* : $28$
* 
* 
* 
* 
* 
Category:Are All Perfect Numbers Even?
Category:Perfect Numbers
Category:Open Questions
Are All Perfect Numbers Even?/Progress

== Progress on Open Question: Are All Perfect Numbers Even? ==
While it is not known whether there exist any odd perfect numbers, several important facts have been established.
=== Minimum Size of Odd Perfect Number ===
=== Form of Odd Perfect Number ===
=== Prime Factors of Odd Perfect Number ===
== Sources ==
* : $28$
* : $28$
Category:Are All Perfect Numbers Even?
Are All Perfect Numbers Even?/Progress/Form

== Theorem ==
An odd perfect number $n$ is of the form:
:$n = p^a q^b r^c \cdots$
where:
:$p, q, r, \ldots$ are prime numbers of the form $4 k + 1$ for some $k \in \Z_{>0}$
:$a$ is also of the form $4 k + 1$ for some $k \in \Z_{>0}$
:$b, c, \ldots$ are all even.
== Proof ==
== Historical Note ==
== Sources ==
* : $28$
* : $28$
Category:Are All Perfect Numbers Even?
Are All Perfect Numbers Even?/Progress/Minimum Size

== Theorem ==
It had been established by $1986$ that an odd perfect number, if one were to exist, would have over $200$ digits.
By $1997$ that lower bound had been raised to $300$ digits.
By $2012$ that lower bound had been raised again to $1500$ digits.
== Proof ==
== Historical Note ==
== Sources ==
* : $28$
* : $28$
Category:Are All Perfect Numbers Even?
Are All Perfect Numbers Even?/Progress/Minimum Size/Historical Note

== Historical Note on Minimum Size of Odd Perfect Number ==
 published a proof in $1968$ that an odd perfect number is greater than $10^{36}$.
 published a proof in $1973$ that an odd perfect number is greater than $10^{50}$.
 and published a proof in $1989$ that an odd perfect number is greater than $10^{160}$.
, and published a proof in $1991$ that an odd perfect number is greater than $10^{300}$.
 and published a proof in $2012$ that an odd perfect number is greater than $10^{1500}$.
== Sources ==
* 
* : $\text {3-5}$ The Use of Computers in Number Theory: Conjecture $2$
* 
* : $28$
* 
* 
* 
* 
Category:Historical Notes
Are All Perfect Numbers Even?/Progress/Prime Factors

== Theorem ==
An odd perfect number has:
:at least $8$ distinct prime factors
:at least $11$ distinct prime factors if $3$ is not one of them
:at least $101$ prime factors (not necessarily distinct)
:its greatest prime factor is greater than $1 \, 000 \, 000$
:its second largest prime factor is greater than $1000$
:at least one of the prime powers factoring it is greater than $10^{62}$
:if less than $10^{9118}$ then it is divisible by the $6$th power of some prime.
== Proof ==
== Historical Note ==
== Sources ==
* : $28$
* : $28$
Category:Are All Perfect Numbers Even?
Are All Perfect Numbers Even?/Progress/Prime Factors/Historical Note

== Historical Note on Prime Factors of Odd Perfect Number ==
 stated that there exist no odd perfect number with fewer than $6$ distinct prime factors, and proved that there are none with fewer than $8$ if none of those prime factors is $3$.
 published a proof in $1973$ that an odd perfect number $P$ has the properties that:
:either:
::at least one of the prime powers factoring $P$ is greater than $10^{18}$
::the power of such a prime factor is even
:or:
::there is no divisor of $P$ less than $7$.
 published a proof in $1980$ that an odd perfect number has at least $8$ distinct prime factors.
 and published a proof in $1998$ that an odd perfect number has at least one prime factor which is greater than $1 \, 000 \, 000$.
 and published a proof in $1998$ that:
: at least one of the prime powers factoring an odd perfect number is greater than $10^{62}$
: an odd perfect number has more than $101$ prime factor (not neessarily distinct).
== Sources ==
* : Preface
* 
* 
* 
* 
Category:Historical Notes
Are All Triperfect Numbers Even?/Progress

== Progress on Open Question: Are All Triperfect Numbers Even? ==
While it is not known whether there exist any odd triperfect numbers, several important facts have been established.
=== Minimum Size of Odd Triperfect Number ===
=== Form of Odd Triperfect Number ===
=== Prime Factors of Odd Triperfect Number ===
== Sources ==
* 
* : $120$
* : $120$
Category:Are All Triperfect Numbers Even?
Are All Triperfect Numbers Even?/Progress/Minimum Size

== Theorem ==
It has been established that an odd triperfect number, if one were to exist, would be greater than $10^{70}$.
If it does not have $3$ as a prime factor, then it is greater than $10^{108}$.
== Proof ==
== Sources ==
* 
* : $120$
* : $120$
Category:Are All Triperfect Numbers Even?
Are Irrational Square Roots Normal?

== Open Question ==
It is not known whether irrational square roots are normal.
== Progress ==
It has been established by testing the distribution of the digits of the square roots of the integers $2$ to $15$, excluding $4$ and $9$, in bases $2$, $4$, $8$ and $16$, that they may well be normal.
== Sources ==
* 
:::: is mistakenly reported as Beyler in .
* : $0 \cdotp 12345 67891 01112 13141 51617 18192 02122 \ldots$
* : $0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$
Category:Open Questions
Category:Normal Numbers
Are there Any More Unitary Perfect Numbers?

== Open Question ==
Apart from the ones which are known about, are there any more unitary perfect numbers?
From Sequence of Unitary Perfect Numbers:
== Progress ==
It is known that any larger unitary perfect number must have the following properties:
:It must have at least $8$ prime factors
:It must be divisible by a power of $10$ greater than $2^{10}$.
== Sources ==
* 
* : $87,360$
Category:Open Questions
Category:Unitary Perfect Numbers
Are there Integers which are Sum of 2 Fifth Powers in 2 Ways?

== Open Question ==
It is not known whether there exists an integer which equals the sum of two fifth powers in $2$ different ways.
That is, it is not known whether the Diophantine equation:
:$a^5 + b^5 = c^5 + d^5$
has a solution.
== Progress ==
It is known that no such solutions exist for sums up to $1 \cdotp 02 \times 10^{26}$.
== Sources ==
* : $1,375,298,099$
* 
* : $1,375,298,099$
Category:Fifth Powers
Category:Open Questions
Are there any Palindromic Fourth Powers with Non-Palindromic Roots?

== Open Question ==
It is not known whether there exist any palindromic fourth powers of a non-palindromic integer.
== Progress ==
All known palindromic fourth powers are of palindromic integers, for example:
:$11^4 = 14 \, 641$
== Sources ==
* : $10,662,526,601$
* : $10,662,526,601$
Category:Fourth Powers
Category:Palindromic Numbers
Category:Open Questions
Area Element/Examples/Surface of Body

== Example of Area Element ==
An area element $\delta \mathbf S$ is often coincident or approximately coincident with part of the surface of a body in space.
Such a body can be considered to have the whole of $S$ covered by such as $\delta \mathbf S$.
The direction of the normal to such a $\delta \mathbf S$ is conventionally taken to be the outward normal of $B$.
If $B$ is complicated in shape, a redefinition of the term outward normal may be appropriate.
== Sources ==
* : Chapter $1$: Force and energy in electrostatics: $1.4$ Gauss's Law: $1.4.2$ The flux of the electric field out of a closed surface
Category:Area Elements
Area Enclosed by First Turn of Archimedean Spiral

== Theorem ==
Let $S$ be the Archimedean spiral defined by the equation:
:$r = a \theta$
The area $\AA$ enclosed by the first turn of $S$ and the polar axis is given by:
:$\AA = \dfrac {4 \pi^3 a^2} 3$
:
== Proof ==
== Historical Note ==
== Sources ==
* : Chapter $\text {A}.5$: Archimedes (ca. B.C.)
Category:Archimedean Spiral
Area Enclosed by First Turn of Archimedean Spiral/Historical Note

== Historical Note on Area Enclosed by First Turn of Archimedean Spiral ==
The Area Enclosed by First Turn of Archimedean Spiral was first determined by in his book .
His proof appears as Proposition $24$.
== Sources ==
* : Chapter $\text {A}.5$: Archimedes (ca. B.C.)
Category:Historical Notes