nyuuzyou/wikis huggingface dataset:
Smallest Field is Field

== Theorem ==
The ring $\struct {\set {0_R, 1_R}, +, \circ}$ is the smallest algebraic structure which is a field.
== Proof ==
From Field Contains at least 2 Elements, a field must contain at least two elements.
Hence the null ring, which contains one element, is not a field.
For $\struct {\set {0_R, 1_R}, +, \circ}$ to be a field:
:$\struct {\set {0_R, 1_R}, +}$ must be an abelian group.
This is fulfilled as this is the parity group.
:$\struct {\set {0_R, 1_R}, \circ}$ must be a commutative division ring.
This is fulfilled, as $\struct {\set {0_R, 1_R}^*, \circ} = \struct {\set {1_R}, \circ}$ is the trivial group.
:$\circ$ needs to distribute over $+$. This follows directly from Ring Product with Zero and the behaviour of the identity element in a group.
== Sources ==
* : Chapter $1$: Integral Domains: $\S 5$. Further Examples of Integral Domains: Example $6$ (which proves the long way round that this is an integral domain)
* : Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Example $4$
* : $\S 2.4$: The rational numbers and some finite fields
* 
* 
Category:Field Theory
Smallest Fifth Power which is Sum of 6 Fifth Powers

== Theorem ==
The smallest fifth power which is the sum of $6$ fifth powers is $12^5 = 248 \, 832$:
:$12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5$
== Proof ==
We have:
== Sources ==
* : $248,832$
* : $248,832$
Category:Fifth Powers
Smallest Fourth Power as Sum and Difference of Fourth Powers

== Theorem ==
The smallest $4$th power that can be expressed as the sum of $2$ $4$th powers minus a $3$rd is:
:$2401 = 7^4 = 227^4 + 157^4 - 239^4$
with all numbers less than $10^4$.
== Proof ==
== Sources ==
* 
* : $2401$
Category:Fourth Powers
Category:2401
Smallest Fourth Power as Sum of 5 Distinct Fourth Powers

== Theorem ==
The smallest $4$th power which can be expressed as the sum of $5$ distinct $4$th powers is:
:$15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$
== Proof ==
By Fermat's Little Theorem, for $5 \nmid a$:
:$a^4 \equiv 1 \pmod 5$
For $5 \divides a$:
:$a^4 \equiv 0 \pmod 5$
Therefore in order for the equality to hold, one of the following must be true:
:The $4$th power on the is not divisible by $5$, while exactly $1$ of the $5$ $4$th powers on the is not divisible by $5$
:The $4$th power on the is divisible by $5$, while all of the $5$ $4$th powers on the is divisible by $5$
:The $4$th power on the is divisible by $5$, while none of the $5$ $4$th powers on the is divisible by $5$.
In the first two cases, there must be at least $4$ distinct $4$th powers divisible by $5$.
The smallest multiples of $5$ are $0, 5, 10, 15$.
Hence the $4$th powers on the must exceed $15^4$.
Thus these cases cannot produce a smaller example.
This leaves us with the third case, where is divisible by $5$.
Since $5^4 \ne 4^4 + 3^4 + 2^4 + 1^4 + 0^4$, we only need to check that $10^4$ is not a sum of $5$ distinct $4$th powers.
As $8^4 + 7^4 + 6^4 + 5^4 + 4^4 = 8674 < 10^4$, we must have $9^4$ on the .
As $9^4 + 8^4 = 10657 > 10^4$, we cannot have $8^4$ on the .
As $9^4 + 6^4 + 5^4 + 4^4 + 3^4 = 8819 < 10^4$, we must have $7^4$ on the .
As $9^4 + 7^4 + 6^4 = 10258 > 10^4$, we cannot have $6^4$ on the .
But $9^4 + 7^4 + 5^4 + 4^4 + 3^4 = 9924 < 10^4$.
Therefore $10^4$ is not a sum of $5$ distinct $4$th powers.
This proves that $15^4$ is the smallest $4$th power which can be expressed as the sum of $5$ distinct $4$th powers.
== Also see ==
* Smallest Fourth Power which is Sum of 5 Fourth Powers
== Sources ==
* : $50,625$
* : $50,625$
Category:50,625
Category:15
Category:Fourth Powers
Smallest Fourth Power as Sum of 5 Distinct Fourth Powers/Mistake

== Source Work ==
:
:The Dictionary
::$50,625$
:
:The Dictionary
::$50,625$
== Mistake ==
:Equal to $15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$.
:This is the smallest example of a $4$th power equal to the sum of only $5$ other $4$th powers.
== Correction ==
Should read:
:This is the smallest example of a $4$th power equal to the sum of only $5$ distinct other $4$th powers.
== Also see ==
* Smallest Fourth Power which is Sum of 5 Fourth Powers
== Sources ==
* : $50,625$
* : $50,625$
Category:Mistakes/Curious and Interesting Numbers
Smallest Fourth Power expressible as Sum of 4 Fourth Powers

== Theorem ==
$15 \, 527 \, 402 \, 881$ is the smallest fourth power which can be expressed as the sum of $4$ fourth powers:
:$15 \, 527 \, 402 \, 881 = 353^4 = 30^4 + 120^4 + 272^4 + 315^4$
== Proof ==
== Sources ==
* : $15,527,402,881$
* : $15,527,402,881$
Category:Fourth Powers
Category:15,527,402,881
Smallest Fourth Power which is Sum of 5 Fourth Powers

== Theorem ==
$625$ is the smallest fourth power which is the sum of $5$ fourth powers:
:$625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$
== Proof ==
We check that for $n = 2, 3, 4$, $n^4$ is not a sum of $5$ smaller fourth powers.
We have:
:$5 \times 1^4 = 5 < 16 = 2^4$
:$5 \times 2^4 = 80 < 81 = 3^4$
so $2^4, 3^4$ are not sums of $5$ fourth powers.
For $n = 4$:
:$\dfrac {4^4} {3^4} < 4$
so such a sum can include at most $3$ $3^4$'s. 
However:
:$3 \times 3^4 + 2^4 + 1^4 = 260 > 256 = 4^4$
:$3 \times 3^4 + 2 \ \ \times 1^4 = 245 < 256 = 4^4$
:$2 \times 3^4 + 3 \ \ \times 2^4 = 220 < 256 = 4^4$
therefore $4^4$ is not a sum of $5$ smaller fourth powers.
This shows that $5^4$ is the smallest fourth power which is the sum of $5$ fourth powers.
== Also see ==
* Smallest Fourth Power as Sum of 5 Distinct Fourth Powers
== Sources ==
* : $625$
* : $625$
Category:Fourth Powers
Category:625
Smallest Integer Divisible by All Numbers from 1 to 100

== Theorem ==
The smallest positive integer which is divisible by each of the integers from $1$ to $100$ is:
:$69 \, 720 \, 375 \, 229 \, 712 \, 477 \, 164 \, 533 \, 808 \, 935 \, 312 \, 303 \, 556 \, 800$
== Proof ==
Let $N$ be divisible by each of the integers from $1$ to $100$.
Each prime number between $2$ and $97$ must be a divisor of $N$.
Also:
:$2^6 = 64 \divides N$
:$3^4 = 81 \divides N$
:$5^2 = 25 \divides N$
:$7^2 = 49 \divides N$
Every other integer between $1$ and $100$ is the product of a subset of all of these.
Hence by Euclid's Lemma:
== Sources ==
* : $69,720,375,229,712,477,164,533,808,935,312,303,556,800$
Category:Divisors
Category:69,720,375,229,712,477,164,533,808,935,312,303,556,800
Smallest Integer Divisible by All Numbers from 1 to 100/Historical Note

== Historical Note on Smallest Integer Divisible by All Numbers from 1 to 100 ==
According to in his of $1997$, this result is attributed to .
However, it is an unremarkable and easy enough result that one wonders how original it actually is.
== Sources ==
* : $69,720,375,229,712,477,164,533,808,935,312,303,556,800$
Category:Historical Notes
Smallest Integer not Sum of Two Ulam Numbers

== Theorem ==
The smallest integer greater than $1$ which is not the sum of two Ulam numbers is $23$.
== Proof ==
Recall the Ulam numbers:
We have:
Now consider the the difference between $23$ and successive Ulam numbers:
and it is not necessary to go further back.
Hence the result.
== Sources ==
* 
* : $23$
Category:Ulam Numbers
Category:23
Smallest Integer using Three Words in English Description

== Theorem ==
The smallest integer which uses exactly $3$ words in its standard (British) English description is:
:$21 \, 000$: twenty-one thousand
counting hyphenations as separate words.
== Proof ==
All integers up to $100$ (one hundred) use either $1$ or $2$ words:
:one
:sixty
:seventeen
:ninety-eight
All integers of the form $100 n$ for $n = 1, 2, \ldots 9$ use exactly $2$ words:
:one hundred
:seven hundred
:nine hundred
In British English, the technique for describing integers from $101$ to $199$, and $201$ to $299$ and so on, is to use and between the number of hundreds and the rest:
:one hundred and one
:three hundred and thirteen
:four hundred and twenty-six
:seven hundred and seventy
thus using either $4$ or $5$ words.
All integers of the form $1000 n$ for $n = 1, 2, \ldots 10$ use exactly $2$ words:
:two thousand
:five thousand
:eight thousand
:twelve thousand
:nineteen thousand
:twenty thousand
Similarly with hundreds, the technique for describing integers of the form $1000 m + n$ for $1 \le n \le 99$ is to use and between the number of thousands and the rest:
:five thousand and eighteen
:sixteen thousand and forty-eight
:thirty-seven thousand and sixty
thus using either $4$ or $5$ words.
All other numbers between $1100$ and $20 \, 999$ trivially use more than $3$ words:
:four thousand, eight hundred
:sixteen thousand, one hundred and seventy-seven
:twenty thousand, nine hundred and ninety-nine
and so on.
The smallest integer to use exactly $2$ words is $21$:
:twenty-one
Hence:
:twenty-one thousand
== Sources ==
* : $21,000$
* : $21,000$
Category:Recreational Mathematics
Smallest Integer which is Product of 4 Triples all with Same Sum

== Theorem ==
The smallest integer which can be expressed as the product of $4$ different triplets of integers each of which has the same sum is:
== Proof ==
We have:
== Also see ==
* Smallest Number which is Sum of 4 Triples with Equal Products
== Historical Note ==
== Sources ==
* 
* 
* 
* : $25,200$
* : $\text D 16$: Triples with the same sum and same product
Category:Recreational Mathematics
Category:25,200
Smallest Integer which is Product of 4 Triples all with Same Sum/Historical Note

== Historical Note on Smallest Integer which is Product of 4 Triples all with Same Sum ==
 discusses this result in his of $1981$, and carries it forward into later editions.
In his of $2004$, the result is presented as:
:It may be of interest to ask for the smallest sums or products with each multiplicity. For example, for $4$ triples, finds the smallest common sum to be $118$ ... and the smallest common product to be $25200$ ...
However, in the article cited by , which appears in for Feb. $1981$, in fact does no such thing.
Instead, he raises the question for $5$ such triples.
, in his of $1997$, propagates this, accrediting the result to Mauldron, citing that same problem in .
It is also apparent that Mauldron is a misprint for .
== Sources ==
* 
* 
* : $25,200$
* : $\text D 16$: Triples with the same sum and same product
Category:Historical Notes
Smallest Integer which is Product of 4 Triples all with Same Sum/Mistake

== Source Work ==
:
:The Dictionary
::$25,200$
== Mistake ==
:The smallest number which can be written as the product of $4$ triples, each with the same sum ... The common sum is $137$. [Mauldron, AMM v88]
== Correction ==
The citation is incorrect. The mathematician in question is in fact .
It also needs to be noted that the cited result in the was in fact for $5$ such triples, not $4$.
== Sources ==
* 
* 
* : $25,200$
Category:Mistakes/Curious and Interesting Numbers
Smallest Integer which is Sum of 3 Fifth Powers in 2 Ways

== Theorem ==
The smallest positive integer which can be expressed as the sum of $3$ fifth powers in $2$ different ways:
The positive integer $1 \, 375 \, 298 \, 099$ can be expressed as the sum of $3$ fifth powers in $2$ different ways:
== Proof ==
== Historical Note ==
== Sources ==
* : $1,375,298,099$
* : $1,375,298,099$
* 
Category:Fifth Powers
Category:1,375,298,099
Smallest Integer which is Sum of 3 Sixth Powers in 2 Ways

== Theorem ==
The smallest positive integer which can be expressed as the sum of $3$ sixth powers in $2$ different ways is:
Also note that:
== Proof ==
We have that:
== Sources ==
* : $160,426,514$
* : $160,426,514$
Category:Sixth Powers
Category:160,426,514
Smallest Magic Cube is of Order 3

== Theorem ==
Apart from the trivial order $1$ magic cube:
the smallest magic cube is the order $3$ magic cube:
== Proof ==
Suppose there were an order $2$ magic cube.
Take one row of this magic cube.
From Magic Constant of Magic Cube, the row and column total is $9$.
Any row or column with a $1$ in it must therefore also have an $8$ in it.
But there are:
:one row
:one column
both of which have a $1$ in them.
Therefore the $8$ would need to go in $2$ distinct cells.
But $8$ appears in a magic cube exactly once.
Hence there can be no order $2$ magic cube.
Category:Magic Cubes
Smallest Magic Square is of Order 3

== Theorem ==
Apart from the trivial order $1$ magic square:
the smallest magic square is the order $3$ magic square:
== Proof ==
Suppose there were an order $2$ magic square.
The row and column total is $\dfrac {1 + 2 + 3 + 4} 2 = 5$.
Any row or column with a $1$ in it must therefore also have a $4$ in it.
But there are:
:one row
:one column
:one diagonal
all of which have a $1$ in them.
Therefore the $4$ would need to go in all $3$ cells.
But $4$ appears in a magic square exactly once.
Hence there can be no order $2$ magic square.
== Sources ==
* : $3$
* : $3$
Category:Magic Squares
Smallest Multiple of 9 with all Digits Even

== Theorem ==
$288$ is the smallest integer multiple of $9$ all of whose digits are even.
== Proof 1 ==
== Proof 2 ==
== Historical Note ==
== Sources ==
* : $288$
Category:Recreational Mathematics
Category:9
Category:288
Category:Classic Problems
Category:Smallest Multiple of 9 with all Digits Even
Smallest Multiple of 9 with all Digits Even/Proof 1

== Theorem ==
== Proof ==
By the brute force technique:
All integer multiples of $9$ $k \times 9$ for $k = 12$ through to $k = 22$ begin with $1$, and so have at least one odd digit.
Then:
Hence the result.
Category:Smallest Multiple of 9 with all Digits Even
Smallest Multiple of 9 with all Digits Even/Proof 2

== Theorem ==
== Proof ==
Let $n$ be the smallest integer multiple of $9$ all of whose digits are even.
From Divisibility by 9, the digits of $n$ must add to an integer multiple of $9$.
But from Sum of Even Integers is Even, the digits of $n$ must add to an even integer multiple of $9$: $18, 36, 54$, etc.
There is only $1$ integer multiple of $9$ with $2$ digits whose digits add to an even integer multiple of $9$, and that is $99$.
Thus we have to look at $3$-digit integers.
The following sets of $3$ even digits add to $18$:
:$\set {2, 8, 8}$
:$\set {4, 6, 8}$
:$\set {6, 6, 6}$
and that seems to be about it.
There are no sets of $3$ even digits which add up to $36$ or higher.
The result follows by inspection.
Category:Smallest Multiple of 9 with all Digits Even
Smallest Multiplicative Magic Square is of Order 3

== Theorem ==
The order of the smallest multiplicative magic square is $3$, for example:
== Proof ==
Suppose there were an order $2$ multiplicative magic square $M$.
Let $a$ be the element of row $1$ and column $1$.
Let $a b$ be the magic constant of $M$.
Then $b$ is:
:the element of row $1$ and column $2$, to make the product of row $1$ equal to $a b$
:the element of row $2$ and column $1$, to make the product of column $1$ equal to $a b$
and so on.
But it is a principle of multiplicative magic squares that the elements are all distinct.
Hence no order $2$ multiplicative magic square can exist.
Category:Multiplicative Magic Squares
Smallest Multiply Perfect Number of Order 5

== Theorem ==
The number $14 \, 182 \, 439 \, 040$ is multiply perfect of order $5$:
:$\map {\sigma_1} {14 \, 182 \, 439 \, 040} = 70 \, 912 \, 195 \, 200 = 5 \times 14 \, 182 \, 439 \, 040$
It is the smallest positive integer to be so.
== Proof ==
From :
:$\map {\sigma_1} {14 \, 182 \, 439 \, 040} = 70 \, 912 \, 195 \, 200$
== Sources ==
* 
* : $14,182,439,040$
Category:Multiply Perfect Numbers
Category:14,182,439,040
Smallest Multiply Perfect Number of Order 6

== Theorem ==
The number $154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600$ is multiply perfect of order $6$:
It is the smallest positive integer to be so.
== Proof ==
From :
:$\map {\sigma_1} {154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600} = 926 \, 073 \, 336 \, 514 \, 623 \, 897 \, 600$
== Sources ==
* : $154,345,556,085,770,649,600$
Category:Multiply Perfect Numbers
Category:154,345,556,085,770,649,600
Smallest Non-Palindromic Number with Palindromic Square

== Theorem ==
$26$ is the smallest non-palindromic integer whose square is palindromic.
== Proof ==
Checking the squares of all non-palindromic integers in turn from $10$ upwards, until a palindromic integer is reached:
== Sources ==
* : $26$
* : $26$
Category:26
Category:Square Numbers
Category:Palindromic Numbers
Smallest Normal Subgroup containing Set

== Theorem ==
Let $S \subseteq G$ where $G$ is a group.
Then there exists a unique smallest normal subgroup of $G$ which contains $S$.
== Proof ==
Let $\Bbb S$ be the set of all normal subgroups of $G$ that contain $S$.
Since $S \subseteq G \lhd G$, $\Bbb S \ne \O$.
Let $N = \bigcap \Bbb S$, that is, the intersection of all elements of $\Bbb S$.
By Intersection of Normal Subgroups is Normal, $N \lhd G$.
By the definition of intersection, $S \subseteq N$.
By the method of construction, $N$ is the smallest such subgroup.
By the definition of "smallest", $N$ is unique.
Category:Normal Subgroups
Smallest Number Expressible as Sum of at most Three Triangular Numbers in 4 ways

== Theorem ==
$21$ is the smallest number which can be expressed as the sum of at most $3$ triangular numbers in $4$ ways.
== Proof ==
By inspection:
== Sources ==
* 
::in which there are many errata
* 
::in which the errata in the above are resolved
* : $21$
Category:Triangular Numbers
Category:21
Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers

== Theorem ==
The smallest positive integer which cannot be expressed as the sum of fewer than $19$ fourth powers is $79$:
:$79 = 15 \times 1^4 + 4 \times 2^4$
== Proof ==
We have $1^4 = 1, 2^4 = 16, 3^4 = 81 > 79$.
Hence for each $n < 79$, we can only use $1^4$ and $2^4$ in our sum.
Write $n = 2^4 a + 1^4 b$.
We can use the greedy algorithm to generate these expressions, since replacing $2^4$ with $16 \times 1^4$ increases the number of fourth powers required.
Suppose $n < 64$.
By Division Theorem, there is a unique way to write $n = 16 q + r$, with $q \in \Z$, $0 \le r < 16$.
Thus $q + r \le 3 + 15 = 18$.
It follows that each positive integer less than $64$ can be expressed in not more than $18$ fourth powers.
Suppose $64 \le n \le 78$.
We cannot use more than $4$ $2^4$, since $5 \times 2^4 = 80 > n$.
Thus we write $n = 4 \times 2^4 + \paren {n - 64} \times 1^4$.
Since $n - 64 \le 78 - 64 = 14$, we can use not more than $18$ fourth powers to express $n$.
For $n = 79$, we still cannot use more than $4$ $2^4$, since $5 \times 2^4 = 80 > n$.
Thus $n = 4 \times 2^4 + 15 \times 1^4$ uses the least number of fourth powers.
Hence $79$ is the smallest positive integer which cannot be expressed as the sum of fewer than $19$ fourth powers.
== Also see ==
* Hilbert-Waring Theorem
* 159 is not Expressible as Sum of Fewer than 19 Fourth Powers
* 319 is not Expressible as Sum of Fewer than 19 Fourth Powers
== Sources ==
* : $79$
* : $79$
Category:Fourth Powers
Category:Hilbert-Waring Theorem
Category:79
Smallest Number to appear 6 Times in Pascal's Triangle

== Theorem ==
The smallest positive integer greater than $1$ to appear $6$ times in Pascal's Triangle is $120$.
== Proof ==
We have:
:$\dbinom {120} 1 = \dbinom {16} 2 = \dbinom {10} 3 = \dbinom {10} 7 = \dbinom {16} {14} = \dbinom {120} {119} = 120$
To verify that this is the smallest, we look at binomial coefficients that are no more than $120$.
Observe that for $n > 120$, $1 \le k \le n - 1$:
:$\dbinom n k \ge \dbinom n 1 = n > 120$
For $n > 16$, $2 \le k \le n - 2$:
:$\dbinom n k \ge \dbinom n 2 = \dfrac {n \paren {n - 1}} {2!} > \dfrac {16 \paren {15}} {2!} = 120$
For $n > 10$, $3 \le k \le n - 3$:
:$\dbinom n k \ge \dbinom n 3 = \dfrac {n \paren {n - 1} \paren {n - 2}} {3!} > \dfrac {10 \paren 9 \paren 8} {3!} = 120$
Any number, except $2$, appears twice in the form $\dbinom n 1$ and $\dbinom n {n - 1}$.
Therefore we scout for numbers appearing four times, not as those forms, by listing $\dbinom n k$ for:
:$n \le 10$ and $2 \le k \le n - 2$;
:$n \le 16$ and $k = 2$ or $n - 2$
They are:
:$6$
:$10, 10$
:$15, 20, 15$
:$21, 35, 35, 21$
:$28, 56, 70, 56, 28$
:$36, 84, 126, 126, 84, 36$
:$45, 120, *, *, *, 120, 45$
:$55, 55$
:$66, 66$
:$78, 78$
:$91, 91$
:$105, 105$
:$120, 120$
and we see that the only number appearing four times is $120$.
== Sources ==
* : $120$
* : $120$
Category:Pascal's Triangle
Category:120
Smallest Number which is Multiplied by 99 by Appending 1 to Each End

== Theorem ==
The smallest positive integer which is multiplied by $99$ when $1$ is appended to each end is:
:$112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 809$
== Proof ==
We have that:
:$112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 809 = 101 \times 1 \, 052 \, 788 \, 969 \times 1 \, 056 \, 689 \, 261$
while:
Let $N$ be the smallest integer satisfying $99 N = \sqbrk {1N1}$ when expressed in decimal notation.
Suppose $N$ is $k$ digits long.
Then:
:$\sqbrk {1 N 1} = 10^{k + 1} + 10 N + 1$
Subtracting $10 N$ from $99 N$ gives:
:$89 N = 10^{k + 1} + 1$
One can show, by trial and error, that the smallest $k$ where $10^{k + 1} + 1$ is divisible by $89$ is $21$.
Then $N = \dfrac {10^{22} + 1} {89} = 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 809$.
== Sources ==
* 
* : $112,359,550,561,797,732,809$
:::but note the transcription error in the $5$th digit from the end: it should be $5$ not $3$.
Category:Recreational Mathematics
Category:112,359,550,561,797,752,809
Smallest Number which is Multiplied by 99 by Appending 1 to Each End/Mistake

== Source Work ==
:
:The Dictionary
::$112,359,550,561,797,732,809$
== Mistake ==
:The smallest number which, when $1$ is placed at both ends, the number is multiplied by $99$.
== Correction ==
There is a transcription error in the $5$th digit from the end: it should be $5$ not $3$.
The correct number is $112,359,550,561,797,752,809$.
See that:
:$112,359,550,561,797,732,809 \times 99 = 11,123,595,505,617,975,548,091$
== Sources ==
* : $112,359,550,561,797,732,809$
Category:Mistakes/Curious and Interesting Numbers
Smallest Number which is Sum of 4 Triples with Equal Products

== Theorem ==
The smallest positive integer which is the sum of $4$ 
distinct ordered triples, each of which has the same product, is $118$:
== Proof ==
== Computational proof ==
The following python script finds the maximum number of ways to represent each positive integer as sums of triples with the same product:
def SearchTriplets(n):
 rmap = {}
 # iterate i up to i + 1 + 1 = n
 for i in range(1,n-2):
 # iterate j up to i + j + 1 = n
 for j in range(i,n-i-1):
 k = n - i - j
 vec = [i,j,k]
 # sort list to make unique representation
 vec.sort()
 p1 = i*j*k
 # Add or insert vec to product dictionary if it does not exist
 if p1 in rmap:
 if not vec in rmap[p1]:
 rmap[p1].append(vec)
 else:
 rmap[p1] = [ vec ]
 maxv = []
 products = list(rmap.keys())
 # report longest list - with lowest product if multiple list with equal lengths exist
 products.sort()
 for p in products:
 v = rmap[p]
 if len(v) > len(maxv):
 maxv = v
 return maxv
small = 1
for i in range(1000):
 res = SearchTriplets(i)
 if len(res) > small:
 print(i, len(res), res)
 small = len(res)
This outputs:
13 2 [[1, 6, 6], [2, 2, 9]]
39 3 [[4, 15, 20], [5, 10, 24], [6, 8, 25]]
118 4 [[14, 50, 54], [15, 40, 63], [18, 30, 70], [21, 25, 72]]
185 5 [[11, 84, 90], [12, 63, 110], [15, 44, 126], [18, 35, 132], [22, 28, 135]]
400 6 [[24, 180, 196], [27, 128, 245], [28, 120, 252], [32, 98, 270], [36, 84, 280], [42, 70, 288]]
511 7 [[35, 216, 260], [36, 195, 280], [40, 156, 315], [42, 144, 325], [45, 130, 336], [60, 91, 360], [72, 75, 364]]
This confirms the statement that $118$ is the smallest positive integer that satisfies the criterion.
== Also see ==
* Smallest Integer which is Product of 4 Triples all with Same Sum
== Historical Note ==
== Sources ==
* 
* 
* : $118$
* 
* : $118$
* : $\text D 16$: Triples with the same sum and same product
Category:118