nyuuzyou/wikis huggingface dataset:
Laplace Transform of Positive Integer Power

== Theorem ==
Let $\laptrans f$ denote the Laplace transform of a function $f$.
Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$.
Then:
:$\laptrans {t^n} = \dfrac {n!} { s^{n + 1} }$
for $\map \Re s > 0$.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Laplace Transform of Complex Power
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of some Elementary Functions: $3$.
* : $\S 32$: Table of Special Laplace Transforms: $32.27$
Category:Laplace Transforms
Category:Laplace Transform of Positive Integer Power
Laplace Transform of Positive Integer Power/Proof 2

== Theorem ==
== Proof ==
The proof proceeds by induction on $n$ for $t^n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\laptrans {t^n} = \dfrac {n!} { s^{n + 1} }$
=== Basis for the Induction ===
$\map P 0$ is the case:
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
:$\laptrans {t^k} = \dfrac {k!} {s^{k + 1} }$
from which it is to be shown that:
:$\laptrans {t^{k + 1} } = \dfrac {\paren {k + 1}!} {s^{k + 2} }$
=== Induction Step ===
This is our induction step:
From Integration by Parts:
:$\ds \int f g' \rd t = f g - \int f' g \rd t$
Here:
So:
Evaluating at $t = 0$ and $t \to +\infty$:
The result follows by the Principle of Mathematical Induction.
== Sources ==
* 
Category:Laplace Transform of Positive Integer Power
Laplace Transform of Real Power

== Theorem ==
Let $n$ be a constant real number such that $n > -1$
Let $f: \R \to \R$ be the real function defined as:
:$\map f t = t^n$
Then $f$ has a Laplace transform given by:
where $\Gamma$ denotes the gamma function.
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $1$
* : Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $31$
* : $\S 15$: Definite Integrals involving Exponential Functions: $15.76$
* : $\S 32$: Table of Special Laplace Transforms: $32.28$
Category:Laplace Transforms
Category:Definite Integrals involving Exponential Function
Laplace Transform of Reciprocal of Square Root

== Theorem ==
:$\forall t \in \R_{\ne 0}: \laptrans {\dfrac 1 {\sqrt t} } = \sqrt {\dfrac \pi s}$
where $\laptrans f$ denotes the Laplace transform of the real function $f$.
== Proof ==
Let $\map f t = \dfrac 1 {\sqrt t}$.
By definition of the Laplace transform with a discontinuity at zero, $\laptrans f$ is the improper integral:
:$\ds \laptrans f := \lim_{\epsilon \mathop \to 0^+} \int_\epsilon^{\to +\infty} \dfrac {e^{-s t} } {\sqrt t} \rd t$
if it exists.
It follows that this improper integral converges.
So the Laplace transform $\laptrans f$ exists.
Hence:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $32$
Category:Laplace Transforms
Laplace Transform of Second Derivative

== Theorem ==
Let $f: \R \to \R$ or $\R \to \C$ be a continuous function on any interval of the form $0 \le t \le a$.
Let $f$ be twice differentiable.
Let $f'$ be continuous and $f' '$ piecewise continuous with one-sided limits on said intervals.
Let $f$ and $f'$ be of exponential order.
Let $\laptrans f$ denote the Laplace transform of $f$.
Then $\laptrans {f' '}$ exists for $\map \Re s > a$, and:
:$\laptrans {\map {f' '} t} = s^2 \laptrans {\map f t} - s \, \map f 0 - \map {f'} 0$
== Proof ==
== Also see ==
* Laplace Transform of Higher Order Derivatives
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $5$. Laplace transform of derivatives: Theorem $1 \text{-} 9$
* : Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transform of Derivative: $14$
* : $\S 32$: Table of General Properties of Laplace Transforms: $32.8$
* : $\S 6.2$
* 
Category:Laplace Transforms of Derivatives
Laplace Transform of Shifted Dirac Delta Function

== Theorem ==
Let $\map \delta t$ denote the Dirac delta function.
The Laplace transform of $\map \delta {t - a}$ is given by:
:$\laptrans {\map \delta {t - a} } = e^{-a s}$
== Proof 1 ==
== Also see ==
* Laplace Transform of Dirac Delta Function
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $13$
Category:Laplace Transforms
Category:Dirac Delta Function
Category:Laplace Transform of Shifted Dirac Delta Function
Laplace Transform of Sine

== Theorem ==
Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Proof 5 ==
== Also see ==
* Laplace Transform of Cosine
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of some Elementary Functions: $5$.
* : $\S 32$: Table of Special Laplace Transforms: $32.32$
Category:Laplace Transforms‎
Category:Sine Function
Category:Laplace Transform of Sine
Laplace Transform of Sine/Proof 4

== Theorem ==
== Proof ==
By definition of the Laplace transform:
:$\ds \laptrans {\sin a t} = \int_0^{\to +\infty} e^{-s t} \sin a t \rd t$
From Integration by Parts:
:$\ds \int f g' \rd t = f g - \int f' g \rd t$
Here:
So:
Consider:
:$\ds \int e^{-s t} \cos a t \rd t$
Again, using Integration by Parts:
:$\ds \int h j\,' \rd t = h j - \int h' j \rd t$
Here:
So:
Substituting this into $(1)$:
Evaluating at $t = 0$ and $t \to +\infty$:
== Sources ==
* 
Category:Laplace Transform of Sine
Laplace Transform of Sine/Proof 5

== Theorem ==
== Proof ==
From Laplace Transform of Second Derivative:
:$(1): \quad \laptrans {\map {f' '} t} = s^2 \laptrans {\map f t} - s \, \map f 0 - \map {f'} 0$
under suitable conditions.
Then:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transform of Derivative: $16$
Category:Laplace Transform of Sine
Laplace Transform of Sine Integral Function

== Theorem ==
:$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\Si$ denotes the sine integral function
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $8$
Category:Laplace Transforms
Category:Sine Integral Function
Category:Laplace Transform of Sine Integral Function
Laplace Transform of Sine Integral Function/Proof 1

== Theorem ==
== Proof ==
Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
By the Initial Value Theorem of Laplace Transform:
:$\ds \lim_{s \mathop \to \infty} s \laptrans {\map f t} = \lim_{t \mathop \to 0} \map f t = \map f 0 = 0$
which leads to:
:$c = \dfrac \pi 2$
Thus:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: The Sine, Cosine and Exponential Integrals: $36$: Method $1$
Category:Laplace Transform of Sine Integral Function
Laplace Transform of Sine Integral Function/Proof 3

== Theorem ==
== Proof ==
Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: The Sine, Cosine and Exponential Integrals: $36$: Method $3$
Category:Laplace Transform of Sine Integral Function
Laplace Transform of Sine Integral Function/Proof 4

== Theorem ==
== Proof ==
Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: The Sine, Cosine and Exponential Integrals: $36$: Method $4$
Category:Laplace Transform of Sine Integral Function
Laplace Transform of Sine of Root

== Theorem ==
:$\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$
where $\laptrans f$ denotes the Laplace transform of the function $f$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $4$
Category:Laplace Transforms
Category:Sine Function
Category:Laplace Transform of Sine of Root
Laplace Transform of Sine of Root/Proof 2

== Theorem ==
== Proof ==
Let $\map y t := \sin \sqrt t$.
Differentiating twice $t$, we get:
:$(1): \quad 4 t y + 2 y + y = 0$
Let $\map Y s = \laptrans {\map t y}$ be the Laplace transform of $y$.
Then taking the Laplace transform of $(1)$:
For small $t$, we have:
:$\sin \sqrt t \sim \sqrt t$
and:
:$\laptrans {\sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} }$
For large $t$, we have:
:$Y \sim \dfrac c {2 s^{3/2} }$
Hence by comparison:
:$c = \dfrac {\sqrt \pi} 2$
Hence:
:$\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Miscellaneous Problems: $48$
Category:Laplace Transform of Sine of Root
Laplace Transform of Sine of t over t

== Theorem ==
Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\dfrac {\sin t} t} = \arctan \dfrac 1 s$
=== Corollary ===
== Proof ==
From Limit of $\dfrac {\sin x} x$ at Zero:
:$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$
From Laplace Transform of Sine:
:$(1): \quad \laptrans {\sin t} = \dfrac 1 {s^2 + 1}$
From Laplace Transform of Integral:
:$(2): \quad \ds \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$
Hence:
== Also see ==
* Laplace Transform of Sine Integral Function
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $8$. Division by $t$
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $12$
Category:Laplace Transforms‎
Category:Sine Function
Category:Laplace Transform of Sine
Laplace Transform of Sine of t over t/Corollary

== Theorem ==
Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\dfrac {\sin a t} t} = \arctan \dfrac a s$
== Proof ==
But:
The result follows.
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $12$
Category:Laplace Transforms‎
Category:Sine Function
Category:Laplace Transform of Sine
Laplace Transform of t^2 by Cosine a t

== Theorem ==
Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {t^2 \cos a t} = \dfrac {2 s^3 - 6 a^2 s} {\paren {s^2 + a^2}^3}$
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Multiplication by Powers of $t$: $20 \ \text{(b)}$
* : $\S 32$: Table of Special Laplace Transforms: $32.59$
Category:Laplace Transforms‎
Category:Cosine Function
Laplace Transform of t by Sine a t

== Theorem ==
Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {t \sin a t} = \dfrac {2 a s} {\paren {s^2 + a^2}^2}$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 32$: Table of Special Laplace Transforms: $32.43$
Category:Laplace Transforms‎
Category:Sine Function
Category:Laplace Transform of t by Sine a t
Laplacian of Function in Orthogonal Curvilinear Coordinates

== Theorem ==
Let $\map \psi {q_1, q_2, q_3}$ denote a real-valued function embedded in an orthogonal curvilinear coordinate system.
Then the Laplacian of $\psi$ can be expressed as:
:$\nabla^2 \psi = \dfrac 1 {h_1 h_2 h_3} \paren {\map {\dfrac \partial {\partial q_1} } {\dfrac {h_2 h_3} {h_1} \dfrac {\partial \psi} {\partial q_1} } + \map {\dfrac \partial {\partial q_2} } {\dfrac {h_3 h_1} {h_2} \dfrac {\partial \psi} {\partial q_2} } + \map {\dfrac \partial {\partial q_3} } {\dfrac {h_1 h_2} {h_3} \dfrac {\partial \psi} {\partial q_3} } }$
where:
:${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$
== Proof ==
== Sources ==
* : Chapter $\text I$: Introduction: $\S 1$. The origin of special functions: $(1.5)$
* : Appendix $16$: Vector algebra and operators
Category:Orthogonal Curvilinear Coordinates
Laplacian on Scalar Field is Divergence of Gradient

== Theorem ==
Let $\R^n$ denote the real Cartesian space of $n$ dimensions.
Let $U$ be a scalar field over $\R^n$.
Let $\nabla^2 U$ denote the laplacian on $U$.
Then:
:$\nabla^2 U = \operatorname {div} \grad U$
where:
:$\operatorname {div}$ denotes the divergence operator
:$\grad$ denotes the gradient operator.
== Proof ==
From Divergence Operator on Vector Space is Dot Product of Del Operator and definition of the gradient operator:
where $\nabla$ denotes the del operator.
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.
Hence:
== Also presented as ==
In Cartesian $3$-space $\R^3$, where:
:$U$ is defined as $\map U {x, y, z}$
:$\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis on $\R^3$.
this is usually presented as:
== Sources ==
* : Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $1$. The Operator $\operatorname {div} \grad$: $(5.1)$, $(5.2)$
* : $\S 22$: The Laplacian: $22.32$
Category:Laplacian
Category:Divergence Operator
Category:Gradient Operator
Largest 9-Digit Prime Number

== Theorem ==
The largest prime number with $9$ digits is $999 \, 999 \, 937$.
== Proof ==
Consider the numbers $\sqbrk {999 \, 999 \, 9ab}$.
Since $999 \, 999 \, 000$ is divisible by $2, 3, 5, 7, 11, 13$,
if $\sqbrk {9ab}$ is divisible by these primes, so is $\sqbrk {999 \, 999 \, 9ab}$.
After this elimination the only $\sqbrk {ab} > 37$ that remains are:
:$41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97$
We have:
And we do have $999 \, 999 \, 937$ is prime.
== Sources ==
* : $999,999,937$
Category:Specific Numbers
Category:999,999,937
Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers

== Theorem ==
Let $k \in \Z_{>0}$ be a (strictly) positive integer.
The largest even integer which cannot be expressed as the sum of $2 k$ odd positive composite integers is $18 k + 20$.
== Proof ==
Let $n$ be an even integer greater than $18 k + 20$.
Then $n - 9 \paren {2 k - 2}$ is an even integer greater than $18 k + 20 - 9 \paren {2 k - 2} = 38$.
By Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers, every even integer greater than $38$ can be expressed as the sum of $2$ odd positive composite integers.
Thus $n - 9 \paren {2 k - 2}$ can be expressed as the sum of $2$ odd positive composite integers.
So, let $a$ and $b$ be odd composite integers such that $a + b = n - 9 \paren {2 k - 2}$.
Then:
:$9 \paren {2 k - 2} + a + b = n$
This is an expression for $n$ as the sum of $2 k$ odd positive composite integers, in which $2 k - 2$ of them are occurrences of $9$.
Thus such an expression can always be found for $n > 18 k + 20$.
It remains to be shown that $18 k + 20$ is not expressible as the sum of $2 k$ odd positive composite integers.
The $k = 1$ case is demonstrated in Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers.
Now suppose $k \ge 2$. 
The two smallest odd positive composite integers are $9$ and $15$.
Suppose $18 k + 20$ is expressible as the sum of $2 k$ odd positive composite integers.
Then at least $2 k - 3$ of them are $9$'s, because:
:$9 \paren {2 k - 4} + 4 \times 15 = 18 k + 24 > 18 k + 20$
Then the problem reduces to finding an expression of $18 k + 20 - 9 \paren {2 k - 3} = 47$ as the sum of $3$ odd positive composite integers.
The first few odd positive composite integers are:
:$9, 15, 21, 25, 27, 33, 35, 39, 45$
Their differences with $47$ are:
:$38, 32, 26, 22, 20, 14, 12, 8, 2$
The integers above are in the set of integers not expressible as a sum of $2$ odd positive composite integers.
The full set of these can be found in Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers.
Thus $18 k + 20$ is not expressible as the sum of $2 k$ odd positive composite integers.
This proves the result.
== Historical Note ==
== Sources ==
* 
* : $38$
Category:Composite Numbers
Category:Odd Integers
Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers/Historical Note

== Historical Note on Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways ==
According to the footnote to the presentation of the solution to Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers in , this particular result was deduced by the Shippensburg University Mathematical Problem Solving Group.
== Sources ==
* 
Category:Historical Notes
Largest Integer Expressible by 3 Digits

== Theorem ==
The largest integer that can be represented using no more than $3$ digits, with no additional symbols, is:
:$9^{9^9} = 9^{387 \, 420 \, 489}$
and (at $369 \, 693 \, 100$ digits, is too large to be calculated on a conventional calculator.
Note that this does not include the notation for tetration: ${}^9 9$.
=== Logarithm Base 10 ===
=== Number of Digits ===
== Sources ==
* : $9^{9^9}$
* : $9^{9^9}$
Category:Largest Integer Expressible by 3 Digits
Category:Powers
Largest Integer Expressible by 3 Digits/Logarithm Base 10

== Theorem ==
:$\map {\log_{10} } {9^{9^9} } \approx 369 \, 693 \,099 \cdotp 63157 \, 03685 \, 87876 \, 1$
== Proof ==
== Historical Note ==
== Sources ==
* : $9^{9^9}$
* : $9^{9^9}$
Category:Largest Integer Expressible by 3 Digits
Largest Integer Expressible by 3 Digits/Logarithm Base 10/Historical Note

== Historical Note on Largest Integer Expressible by 3 Digits: Logarithm Base $10$ ==
 published the value of $\map {\log_{10} } {9^{9^9} }$ to $250$ decimal places in $1947$.
Apparently he found this sort of calculation relaxing.
== Sources ==
* 
* : $9^{9^9}$
* : $9^{9^9}$
Category:Historical Notes
Largest Integer Expressible by 3 Digits/Number of Digits

== Theorem ==
:$9^{9^9}$ has $369 \, 693 \, 100$ digits when expressed in decimal notation.
== Proof ==
Let $n$ be the number of digits in $9^{9^9}$
From Number of Digits in Number:
:$n = 1 + \floor {\map {\log_{10} } {9^{9^9} } }$
where $\floor {\ldots}$ denotes the floor function.
Then:
== Historical Note ==
== Sources ==
* : $9^{9^9}$
* : $9^{9^9}$
Category:Largest Integer Expressible by 3 Digits
Largest Integer Expressible by 3 Digits/Number of Digits/Historical Note

== Historical Note on Largest Integer Expressible by 3 Digits: Number of Digits ==
The number of digits in $9^{9^9}$ was demonstrated to be $369 \, 693 \, 100$ in $1906$ by .
== Sources ==
* : $9^{9^9}$
* : $9^{9^9}$
Category:Historical Notes
Largest Integer not Sum of Two Abundant Numbers

== Theorem ==
The largest integer which is not the sum of $2$ abundant numbers is $20 \, 161$.
== Proof ==
First we show that for $1 < k < 90$, $315 k$ is abundant.
If $k$ is divisible by $3, 5, 7$, note that:
:$945, 1575, 2205$
are all abundant, and $315 k$ is a multiple of at least one of them.
Hence $315 k$ is abundant by Multiple of Abundant Number is Abundant.
If $k$ is not divisible by $3, 5, 7$:
Let $p$ be a prime such that $p \divides k$.
Then:
hence $315 p$ and $315 k$ are abundant.
Since $88$ and $315$ are coprime:
:$88 = 2^3 \times 11$
:$315 = 3^2 \times 5 \times 7$
By Largest Number not Expressible as Sum of Multiples of Coprime Integers, all numbers greater than or equal to:
:$\paren {88 - 1} \paren {315 - 1} = 27 \, 318$
can be expressed as a sum of multiples of $88$ and $315$.
Hence for $n \ge 27 \, 318 + 315 \times 2 = 27 \, 948$:
:$\exists s, t \in \N: 90 > t \ge 2: n = 88 s + 315 t$
and both $88 s$ and $315 t$ are abundant for $s > 0$.
For $s = 0$, $t \ge \dfrac {27 \, 948} {315} > 7 = \paren {2 - 1} \paren {3 - 1} + 5$.
By Largest Number not Expressible as Sum of Multiples of Coprime Integers, $t - 5$ can be expressed as a sum of multiples of $2$ and $3$.
Hence:
:$\exists a, b \in \Z_{> 0}: 2 a + 3 b = t$
This gives:
:$n = 630 a + 945 b$
and both $630 a$ and $945 b$ are abundant.
We still need to find representations for $20 \, 162 < n < 27 \, 948$.
We can check this via brute force.
Using Largest Number not Expressible as Sum of Multiples of Coprime Integers/Generalization, we can narrow down our search to numbers that are not divisible by small primes:
Since $\gcd \set {18, 20} = 2$, the largest multiple of $2$ not expressible as a sum of multiples of $18$ and $20$ is:
:$\dfrac {18 \times 20} 2 - 18 - 20 = 142 < 20161$
Since $\gcd \set {12, 945} = 3$, the largest multiple of $3$ not expressible as a sum of multiples of $12$ and $945$ is:
:$\dfrac {12 \times 945} 3 - 12 - 945 = 2823 < 20161$
Since $\gcd \set {20, 945} = 5$, the largest multiple of $5$ not expressible as a sum of multiples of $20$ and $945$ is:
:$\dfrac {20 \times 945} 5 - 20 - 945 = 2815 < 20161$
Since $\gcd \set {56, 945} = 7$, the largest multiple of $7$ not expressible as a sum of multiples of $56$ and $945$ is:
:$\dfrac {56 \times 945} 7 - 56 - 945 = 6559 < 20161$
All numbers involved above are abundant.
Hence we only need to consider $n$ not divisible by $2, 3, 5, 7$.
== Also see ==
* Even Integers not Sum of Two Abundant Numbers
== Sources ==
* 
* 
* : $20,161$
* : $20,161$
Category:Abundant Numbers
Largest Integer whose Digits taken in Pairs all form Distinct Primes

== Theorem ==
The largest integer which has the property that every pair of its digits taken together is a distinct prime number is $619 \, 737 \, 131 \, 179$.
== Proof ==
Let $p$ be such an integer.
Apart from the first digit, each of its digits forms the second digit of a $2$-digit prime number.
Thus, apart from the first digit, $p$ cannot contain $2$, $4$, $5$, $6$, $8$ or $0$.
$0$ cannot of course be the first digit either.
Thus, apart from the first pair of its digits, the $2$-digit prime numbers contained in $p$ can only ever be in the set:
:$S = \left\{ {11, 13, 17, 19, 31, 37, 71, 73, 79, 97}\right\}$
Let it be assumed that the largest such $p$ contains all of the above.
There are $10$ elements of $S$.
Each of the elements of $S$ apart from one shares both digits with one other element of $S$, apart from the final one in $p$.
Thus the elements of $S$ contribute $11$ digits of $p$ in total.
The first digit of the first pair contributes an extra $1$ digit to $p$.
Thus $p$ contains $12$ digits.
Only $1$ element of $S$ begins with $9$, but $2$ of them end with $9$.
Therefore $9$ must be the last digit of $p$.
Only $3$ elements of $S$ end in $1$, but $4$ of them begin with $1$.
Thus the first pair of digits of $p$ must be a prime number which does not appear in $S$ whose $2$nd digit is $1$.
Thus the first $2$ digits of $p$ can only be $41$ or $61$.
Let us construct $p$ by starting from the left and use the greedy algorithm consisting of select the largest digit available.
:$61$ is larger than $41$, so $p$ can be assumed to start with $61$.
Using this algorithm, we select the elements of $S$ in the following order:
:$61$, $19$, $97$ (which is forced), $73$, $37$, $71$
The next element of $S$ cannot be $17$, because that would force $79$ to follow.
But as $79$ is the only remaining element of $S$ ending in $9$, $79$ must be at the end of $p$.
So $71$ must be followed by $11$ or $13$.
Continuing to use the greedy algorithm:
:$61$, $19$, $97$ $73$, $37$, $71$, $13$, $31$ (forced)
For the same reason as above, we cannot then select $17$, as this will be followed by $79$ which cannot be followed.
So we continue with the greedy algorithm:
:$61$, $19$, $97$ $73$, $37$, $71$, $13$, $31$, $11$, $17$, $79$
and $p$ is complete:
:$619 \, 737 \, 131 \, 179$
As a bonus, $p$ is itself a prime number.
== Historical Note ==
== Sources ==
* : $619,737,131,179$
* : $619,737,131,179$
Category:Prime Numbers
Category:Recreational Mathematics
Largest Integer whose Digits taken in Pairs all form Distinct Primes/Historical Note

== Historical Note on Largest Integer whose Digits taken in Pairs all form Distinct Primes ==
 reports in his of $1997$ that this factoid appears in issue $40$ of , but this has not been corroborated.
== Sources ==
* : $619,737,131,179$
* : $619,737,131,179$
Category:Historical Notes