nyuuzyou/wikis huggingface dataset:
Smallest Number which is Sum of 4 Triples with Equal Products/Historical Note

== Historical Note on Smallest Number which is Sum of 4 Triples with Equal Products ==
 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 ==
* 
* 
* : $118$
* : $\text D 16$: Triples with the same sum and same product
Category:Historical Notes
Smallest Number which is Sum of 4 Triples with Equal Products/Mistake

== Source Work ==
:
:The Dictionary
::$118$
== Mistake ==
:$118$ is the smallest number which can be written as the sum of four triples, whose products are all equal: $118 = 14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 = 72$.
:The product of each triple is $37,800$. [Mauldron, Guy, 172]
== Correction ==
That first name should read Mauldon.
It appears that this most probably refers to , who published similar results for $5$ such triples as a problem in in $1981$.
== Sources ==
* 
* 
* : $118$
Category:Mistakes/Curious and Interesting Numbers
Smallest Number with 16 Divisors

== Theorem ==
The smallest positive integer with $16$ divisors is $120$.
== Proof ==
From :
:$\map {\sigma_0} {120} = 16$
The result is a specific instance of Smallest Number with $2^n$ Divisors:
:$120 = 2 \times 3 \times 4 \times 5$
== Sources ==
* : $120$
* : $120$
Category:Divisor Count Function
Category:120
Smallest Number with 2^n Divisors

== Theorem ==
The smallest positive integer with $2^n$ divisors is found by multiplying together the first $n$ numbers in this sequence:
:$2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, \ldots$
which consists of all the positive integers of the form $p^{\paren {2^k} }$ where $p$ is prime and $k \ge 0$.
== Proof ==
== Sources ==
* : $120$
Category:Divisor Count Function
Category:120
Smallest Number with 2^n Divisors/Mistake

== Source Work ==
:
:The Dictionary
::$120$
== Mistake ==
:The smallest number having $2^n$ divisors is found by multiplying together the first $n$ numbers in this sequence: $2$, $3$, $4$, $5$, $7$, $9$, $11$, $13$, $16$, $17$, $19$, $\ldots$ which consists of all the primes and powers of primes.
== Correction ==
The last clause should say:
:... which consists of all the numbers of the form $p^{\paren {2^k} }$ where $p$ is prime and $k \ge 0$.
== Sources ==
* : $120$
Category:Mistakes/Curious and Interesting Numbers
Smallest Numbers with 240 Divisors

== Theorem ==
The smallest integers with $240$ divisors are:
:$720 \, 720, 831 \, 600, 942 \, 480, 982 \, 800, 997 \, 920, \ldots$
== Proof ==
In the below, $\map {\sigma_0} n$ denotes the divisor count function of $n$.
Then:
== Sources ==
* : $240$
* : $240$
Category:Divisor Count Function
Smallest Odd Abundant Number not Divisible by 3

== Theorem ==
The smallest odd abundant number not divisible by $3$ is $5 \, 391 \, 411 \, 025$.
== Proof ==
We have:
:$5 \, 391 \, 411 \, 025 = 5^2 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29$
showing it is not divisible by $3$.
Then from we have:
:$\map {\sigma_1} {5 \, 391 \, 411 \, 025} = 10 \, 799 \, 308 \, 800 = 2 \times 5 \, 391 \, 411 \, 025 + 16 \, 486 \, 750$
demonstrating that $5 \, 391 \, 411 \, 025$ is abundant.
== Sources ==
* 
* : $5,391,411,025$
Category:Abundant Numbers
Category:5,391,411,025
Smallest Odd Number not of form 2 a squared plus p

== Theorem ==
$17$ is the smallest odd number $n$ greater than $3$ which cannot be expressed in the form:
:$n = 2 a^2 + p$
where:
:$p$ is prime
:$a \in \Z_{>0}$ is a (strictly) positive integer.
== Proof ==
First note that $3$ is not so expressible:
and so $3$ cannot be written in such a form.
Then we have:
However, we have:
== Also see ==
* Definition:Stern Prime
== Historical Note ==
== Sources ==
* 
* 
* : $17$
::except that in his statement of the problem, he omits to mention $1$ and $3$.
Category:17
Category:Prime Numbers
Category:Polynomial Expressions for Primes
Category:Goldbach's Lesser Conjecture
Smallest Odd Number not of form 2 a squared plus p/Historical Note

== Historical Note on Smallest Odd Number not of form 2 a squared plus p ==
The question of odd integers expressible in the form $2 a^2 + p$, for $a \ge 0$ and $p$ prime, was initially raised by in a letter to dated $18$ November $1752$, in which he conjectured that all odd integers were so expressible.
At that time, $1$ was considered to be prime. Thus $1 = 2 \times 0^2 + 1$ and $3 = 2 \times 1^2 + 1$ were considered to fit the criteria, as was $17 = 0^2 + 17$.
The conjecture was believed to hold until $1856$, when and his students tested all the primes to $9000$, and found the counterexamples $5777$ and $5993$.
He then went on to investigate odd numbers, and more specifically prime numbers, that cannot be represented in the form $2 a^2 + p$ where $a > 0$.
Seeming to forget about $3$, he stated that the smallest such prime number was $17$.
== Sources ==
* 
* 
Category:Historical Notes
Smallest Odd Number not of form 2 a squared plus p/Mistake

== Source Work ==
:
:The Dictionary
::$17$
== Mistake ==
:[$17$ is] the smallest odd number which cannot be represented as the sum of a prime and twice a square.
This is incorrect, as both $1$ and $3$ share that property.
== Sources ==
* : $17$
Category:Mistakes/Curious and Interesting Numbers
Smallest Pair of Quasiamicable Numbers

== Theorem ==
The smallest pair of quasiamicable numbers is $48$ and $75$.
== Proof ==
From Quasiamicable Numbers: $48$ and $75$ we have that $48$ and $75$ are quasiamicable numbers.
It remains to be demonstrated that these are indeed the smallest such.
Let $n$ be the smaller number of the quasiamicable pair.
Then we must have $\map {\sigma_1} n - n - 1 > n$.
Since $\map {\sigma_1} n > 2 n$, $n$ is abundant.
The abundant numbers for $n < 48$ are:
:$12, 18, 20, 24, 30, 36, 40, 42$
And we have:
hence $n$ is at least $48$, proving the result.
== Sources ==
* : $48$
Category:Quasiamicable Numbers
Category:48
Category:75
Smallest Pandigital Square

== Theorem ==
The smallest pandigital square is $1 \, 026 \, 753 \, 849$:
:$1 \, 026 \, 753 \, 849 = 32 \, 043^2$
== Proof ==
We check all the squares of numbers from $\ceiling {\sqrt {1 \, 023 \, 456 \, 789} } = 31 \, 992$ up to $32 \, 042$, with the following constraints:
Since all these squares has $10$ as its two leftmost digits, the number cannot end with $0$, $1$ or $9$.
A pandigital number is divisible by $9$, so our number must be divisible by $3$.
These constraints leaves us with the following $12$ candidates:
By inspection, none of these numbers are pandigital.
== Historical Note ==
== Sources ==
* : $1,026,753,849$
Category:Square Numbers
Category:Pandigital Integers
Category:1,026,753,849
Smallest Penholodigital Square

== Theorem ==
The smallest penholodigital square number is:
:$11 \, 826^2 = 139 \, 854 \, 276$
== Proof ==
Let $n$ be the smallest positive integer whose square is penholodigital.
First it is noted that the smallest penholodigital number is $123 \, 456 \, 789$.
Hence any square penholodigital number must be at least as large as that.
Thus we can can say that:
:$n \ge \ceiling {\sqrt {123 \, 456 \, 789} } = 11 \, 112$
where $\ceiling {\, \cdot \,}$ denotes the ceiling function.
It remains to be demonstrated that no positive integer between $11 \, 112$ and $11 \, 826$ has a penholodigital square.
== Historical Note ==
== Sources ==
* : $11,826$
* : $139,854,276$
* : $11,826$
* : $139,854,276$
Category:Square Numbers
Category:Penholodigital Integers
Category:11,826
Category:139,854,276
Smallest Penholodigital Square/Historical Note

== Historical Note on Smallest Penholodigital Square ==
$139 \, 854 \, 276$ was remarked upon by in his $1716$ work , where he stated:
:This number $139854276$ is a very remarkable Number: First, It's a Square Number; Secondly it contains $9$ Places, and they are $9$ Digits, and I think there is not another that does.
== Sources ==
* : $11,826$
* : $11,826$
Category:Historical Notes
Smallest Penholodigital Square/Mistake

== Source Work ==
:
:The Dictionary
::$11,826$
:
:The Dictionary
::$11,826$
== Mistake ==
:$11,826^2$ is the smallest pandigital square. It was first noted by in $1727$, who thought it was the only pandigital square.
The book in which called attention to this number actually appeared in $1716$.
It was called , and appeared in a number of editions, one of which may well have been published in $1727$. Research is ongoing as to the exact number of editions, and their dates.
Note that in this context pandigital actually means penholodigital: that is, using all the digits except $0$.
== Sources ==
* : $11,826$
* : $11,826$
Category:Mistakes/Curious and Interesting Numbers
Smallest Polyomino with Hole

== Theorem ==
The smallest polyomino with a hole is the heptomino in the form of a $3 \times 3$ 
square with the center $1 \times 1$ square and a corner $1 \times 1$ square missing:
:
== Proof ==
From 35 Hexominoes and by inspection, none of the $35$ hexominoes has a hole.
From Number of Heptominoes and by inspection, exactly one of the $108$ heptominoes has a hole.
This is the smallest polyomino with a hole.
== Sources ==
* : $108$
* : $108$
Category:Polyominoes
Category:Heptominoes
Smallest Positive Integer Combination is Greatest Common Divisor

== Theorem ==
Let $a, b \in \Z_{>0}$ be (strictly) positive integers.
Let $d \in \Z_{>0}$ be the smallest positive integer such that:
: $d = a s + b t$
where $s, t \in \Z$.
Then:
:$(1): \quad d \divides a \land d \divides b$
:$(2): \quad c \divides a \land c \divides b \implies c \divides d$
where $\divides$ denotes divisibility.
That is, $d$ is the greatest common divisor of $a$ and $b$.
== Proof 1 ==
== Proof 2 ==
Category:Greatest Common Divisor
Category:Integer Combinations
Category:Smallest Positive Integer Combination is Greatest Common Divisor
Smallest Positive Integer Combination is Greatest Common Divisor/Proof 1

== Theorem ==
== Proof ==
Let $D$ be the subset of $\Z_{>0}$ defined as:
:$D = \set {a s + b t: s, t \in \Z, a s + b t > 0}$
Setting $s = 1$ and $t = 0$ it is clear that $a = \paren {a \times 1 + b \times 0} \in D$.
So $D \ne \O$.
By Set of Integers Bounded Below by Integer has Smallest Element, $D$ has a smallest element $d$, say.
Thus $d = a s + b t$ for some $s, t \in \Z$.
=== Proof of $(1)$ ===
From the Division Theorem we can write $a = q d + r$ for some $q, r$ with $0 \le r < d$.
If $r \ne 0$ we have:
So by definition of $D$, it is clear that $r \in D$.
But $r < d$ which contradicts the stipulation that $d$ is the smallest element of $D$.
Thus $r = 0$ and so $a = q d$.
That is $d \divides a$.
By the same argument, it follows that $d \divides b$ also.
=== Proof of $(2)$ ===
Let $c \divides a$ and $c \divides b$.
Then:
: $\exists u, v \in \Z: a = c u, b = c v$
Thus:
demonstrating that $c \divides d$.
== Sources ==
* : $\text{A}.1$: Number theory: Theorem $\text{A}.2$
Category:Smallest Positive Integer Combination is Greatest Common Divisor
Smallest Positive Integer Combination is Greatest Common Divisor/Proof 2

== Theorem ==
== Proof ==
From Bézout's Identity we have:
Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.
Then:
:$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
Furthermore, $\gcd \set {a, b}$ is the smallest positive integer combination of $a$ and $b$.
In this instance, $a, b \in \Z_{>0}$ and are therefore both non-zero.
The result then follows by definition of greatest common divisor:
$d = \gcd \set {a, b}$ :
:$(1): \quad d \divides a \land d \divides b$
:$(2): \quad c \divides a \land c \divides b \implies c \divides d$
Category:Smallest Positive Integer Combination is Greatest Common Divisor
Smallest Positive Integer not of form +-4 mod 9 not representable as Sum of Three Cubes

== Conjecture ==
It is possible to express every positive integer, not of the form $\pm 4 \pmod 9$, as the sum of cubes of $3$ integers.
== Progress ==
In $1997$, in his , reported that $30$ was the smallest positive integer that had not been so represented.
Such a representation was found in $1999$:
:$30 = 2220422932^3 + \paren {-2218888517^3} + \paren {-283059965^3}$
As of $2019$, the smallest positive integer for which such a representation has not been found is $114$.
== Sources ==
* 
* : $30$
Category:Cube Numbers
Category:Number Theory
Smallest Positive Integer not of form +-4 mod 9 not representable as Sum of Three Cubes/Mistake

== Source Work ==
:
:The Dictionary
::$30$
== Mistake ==
:$30$ is the smallest number which has not been represented as the sum of $3$ integer cubes.
Integers $n$ of the form $n \equiv \pm 4 \pmod 9$ are not so representable.
The statement needs to be restated as that:
:$30$ is the smallest number not equivalent to $\pm 4 \pmod 9$ which has not been represented as the sum of $3$ integer cubes.
Since the statement was written, $30$ has now been so represented:
:$30 = 2220422932^3 + \left({- 2218888517^3}\right) + \left({- 283059965^3}\right)$
== Sources ==
* : $30$
Category:Mistakes/Curious and Interesting Numbers
Smallest Positive Integer which is Sum of 2 Fourth Powers in 2 Ways

== Theorem ==
The smallest positive integer which can be expressed as the sum of $2$ fourth powers in $2$ different ways is:
== Proof ==
The fact that these are the smallest can be demonstrated by calculation.
== Historical Note ==
== Sources ==
* : $59$
* : $635,318,657$
* : $59$
* : $635,318,657$
* 
Category:Fourth Powers
Category:635,318,657
Smallest Positive Integer which is Sum of 2 Fourth Powers in 2 Ways/Historical Note

== Historical Note on Smallest Positive Integer which is Sum of 2 Fourth Powers in 2 Ways==
This question was raised by in $1772$.
He it was who found the smallest solution, which he did in $1802$.
== Sources ==
* : $59$
* : $635,318,657$
* : $59$
* : $635,318,657$
* 
Category:Historical Notes
Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways

== Theorem ==
The smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$.
== Proof ==
It is determined that there are no smaller numbers with this property by inspection:
From the workings of Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways, $48$ is the smallest positive integer which is the sum of $2$ odd primes in $5$ different ways.
Thus there is no need to explore smaller numbers.
== Sources ==
* : $60$
Category:Prime Numbers
Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways

== Theorem ==
The sequence of positive integers $n$ which are the smallest such that they are the sum of $2$ odd primes in $k$ different ways begins as follows:
:
$k$	$n$
$1$	$6$
$2$	$10$
$3$	$22$
$4$	$34$
$5$	$48$
$6$	$60$
== Proof ==
From Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways, the smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$.
== Sources ==
* : $60$
Category:Prime Numbers
Smallest Prime Number not Difference between Power of 2 and Power of 3

== Theorem ==
$41$ is the smallest prime number which is not the difference between a power of $2$ and a power of $3$.
== Proof ==
First we have:
 $41 = 2^n - 3^m$.
We have that $n > 3$.
Thus:
:$2^n \equiv 0 \pmod 8$
and as:
:$41 \equiv 1 \pmod 8$
we have:
:$1 \equiv -3^m \pmod 8$
which is not possible:
For any integer $k$, 
So for any integer $m$, $1 \not \equiv -3^m \pmod 8$.
Now suppose $41 = 3^m - 2^n$.
We have that $m > 1$ and $n > 2$.
Taking $\mod 3$ on both sides:
which shows that $n$ is even.
Taking $\mod 4$ on both sides:
which shows that $m$ is even as well.
Now we take $\mod 5$ on both sides:
But $\paren {-1}^{m / 2} - \paren {-1}^{n / 2}$ can only be $0$ or $\pm 2$, not $1$, which is a contradiction.
Therefore $41$ is not the difference between a power of $2$ and a power of $3$.
== Sources ==
* 
* 
:::but beware the typo: $31 = 2^4 - 3^0$
* : $41$
Category:Powers of 2
Category:Powers of 3
Category:41
Smallest Prime Number whose Period is of Maximum Length

== Theorem ==
$7$ is the smallest prime number the period of whose reciprocal, when expressed in decimal notation, is maximum:
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$
== Proof ==
From Maximum Period of Reciprocal of Prime, the maximum period of $\dfrac 1 p$ is $p - 1$.
:$\dfrac 1 2 = 0 \cdotp 5$: not recurring.
:$\dfrac 1 3 = 0 \cdotp \dot 3$: recurring with period $1$.
:$\dfrac 1 5 = 0 \cdotp 2$: not recurring.
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$: recurring with period $6$.
== Sources ==
* : $7$
* : $7$
Category:Fractions
Smallest Pythagorean Quadrilateral with Integer Sides

== Theorem ==
The smallest Pythagorean quadrilateral in which the sides of the $4$ right triangles formed by its sides and perpendicular diagonals are all integers has an area of $21 \, 576$.
The sides of the right triangles in question are:
:$25, 60, 65$
:$91, 60, 109$
:$91, 312, 325$
:$25, 312, 313$
== Proof ==
:
The $4$ right triangles are inspected:
The area of each right triangle is calculated:
Thus the total area is:
:$750 + 2730 + 14 \, 196 + 3900 = 21 \, 576$
== Historical Note ==
== Sources ==
* 
* : $21,576$
Category:Pythagorean Triangles
Category:Quadrilaterals
Smallest Pythagorean Quadrilateral with Integer Sides/Historical Note

== Historical Note on Smallest Pythagorean Quadrilateral with Integer Sides ==
 reports in his of $1997$ that this result was published in Volume $21$ no. $9$ by , but this has not been corroborated.
It is also uncertain at this stage exactly what a Pythagorean quadrilateral actually is.
== Sources ==
* : $21,576$
Category:Historical Notes
Smallest Pythagorean Triangle is 3-4-5

== Theorem ==
The smallest Pythagorean triangle has sides of length $3$, $4$ and $5$.
:
== Proof ==
From Solutions of Pythagorean Equation, all Pythagorean triangles, the set of all primitive Pythagorean triples is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where:
:$m, n \in \Z_{>0}$ are (strictly) positive integers
:$m \perp n$, that is, $m$ and $n$ are coprime
:$m$ and $n$ are of opposite parity
:$m > n$.
The smallest two (strictly) positive integers which satisfy the above criteria are:
:$n = 1$
:$m = 2$
Hence:
:$2 m n = 2 \times 2 \times 1 = 4$
:$m^2 - n^2 = 2^2 - 1^2 = 3$
:$m^2 + n^2 = 2^2 + 1^2 = 5$
and to confirm:
:$3^2 + 4^2 = 9 + 16 = 25 = 5^2$
== Sources ==
* : $5$
* : $13$
* : $5$
* : $13$
Category:Pythagorean Triangles
Smallest Quadruplet of Consecutive Integers Divisible by Cube

== Theorem ==
The smallest sequence of quadruplets of consecutive integers each of which is divisible by a cube greater than $1$ is:
:$\tuple {22 \, 624, 22 \, 625, 22 \, 626, 22 \, 627}$
== Proof ==
== Historical Note ==
== Sources ==
* : $1375$
Category:Cube Numbers
Smallest Scalene Obtuse Triangle with Integer Sides and Area

== Theorem ==
The smallest scalene obtuse triangle with integer sides and area has sides of length $4, 13, 15$.
== Proof ==
From Heron's Formula, the area $A$ of $\triangle ABC$ is given by:
:$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
Here we have:
Thus:
== Sources ==
* : $24$
Category:Obtuse Triangles
Category:Scalene Triangles
Smallest Sequence of 5 Consecutive Numbers which are Happy

== Theorem ==
The smallest sequence of $5$ consecutive integers all of which are happy numbers is:
:$44 \, 488, 44 \, 489, 44 \, 490, 44 \, 491, 44 \, 492$
== Proof ==
== Sources ==
* : $44,488$
Category:Happy Numbers
Smallest Set is Unique

== Theorem ==
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$.
Then the smallest set of $\TT$, if it exists, must be unique.
== Proof ==
Let $A, B \in \TT$ both be smallest sets of $\TT$.
Since $A$ is the smallest set:
:$A \subseteq B$
Since $B$ is the smallest set:
:$B \subseteq A$
Hence, by definition of set equality:
:$A = B$
Therefore the smallest set of $\TT$ is unique.
Category:Set Theory
Category:Smallest Elements
Smallest Set may not Exist

== Theorem ==
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$.
The smallest set of $\TT$ may not exist.
== Proof ==
Let $S = \set {0, 1}$ and $\TT = \set {\set 0, \set 1} \in \powerset S$.
Then since $\set 0 \nsubseteq \set 1$:
:$\set 0$ is not the smallest set of $\TT$.
Similarly, since $\set 1 \nsubseteq \set 0$:
:$\set 1$ is not the smallest set of $\TT$.
Therefore $\TT$ has no smallest set.
Category:Set Theory
Category:Smallest Elements
Smallest Set of Weights for One-Pan Balance

== Classic Problem ==
Consider a balance for determining the weight of a physical object.
Let this balance be such that weights may be placed in one of the pans.
What is the smallest set of weights needed to weigh any given integer weight up to a given amount?
== Solution ==
A set of $m$ weights in the sequence $\sequence {2^n}$:
:$1, 2, 4, 8, 16, \ldots$
allows one to weigh any given integer weight up to $2^m - 1$.
== Proof ==
This is equivalent to the statement that every positive integer can be expressed uniquely in binary notation.
This in turn is an application of the Basis Representation Theorem.
== Examples ==
== Also see ==
* Smallest Set of Weights for Two-Pan Balance
== Historical Note ==
== Sources ==
* 
* : $31$
* : Bachet: $108$
* : $31$
Category:Classic Problems
Category:Binary Notation
Category:Smallest Set of Weights for One-Pan Balance
Smallest Set of Weights for One-Pan Balance/Examples/63

== Examples of Smallest Set of Weights for One-Pan Balance ==
Consider a balance such that weights may be placed in one of the pans.
Let $S$ be the smallest set of weights needed to weigh any given integer weight up to $63$ units.
Then $\size S = 6$.
== Proof ==
From Smallest Set of Weights for One-Pan Balance, a set of $6$ weights in the sequence $\sequence {2^n}$:
:$1, 2, 4, 8, 16, 32$
allows one to weigh any given integer weight up to $2^6 - 1 = 63$.
Hence the result.
== Sources ==
* : $\text {1-2}$ The Basis Representation Theorem: Exercise $3$
Category:Smallest Set of Weights for One-Pan Balance
Smallest Set of Weights for One-Pan Balance/Historical Note

== Historical Note on Smallest Set of Weights for One-Pan Balance ==
The question of how many weights are needed to weigh any given integer weight up to a given amount was considered by .
The one-pan balance version of this problem was solved by .
== Sources ==
* : $31$
* : Bachet: $108$
* : $31$
Category:Historical Notes