nyuuzyou/wikis huggingface dataset:
1,234,567,891,234,567,891,234,567,891

== Number ==
$1 \, 234 \, 567 \, 891 \, 234 \, 567 \, 891 \, 234 \, 567 \, 891$ is:
:The $3$rd of the only $3$ known prime numbers consisting of strings of consecutive ascending digits starting from $1$ (allowing either $0$ or $1$ to follow $9$)
== Also see ==
* 
* 
== Sources ==
* : $1,234,567,891$
* : $1,234,567,891$
Category:Specific Numbers
Category:1,234,567,891,234,567,891,234,567,891
1,238,926,361,552,897

== Number ==
$1 \, 238 \, 926 \, 361 \, 552 \, 897$ is:
:The $n$th prime, where $n$ remains to be determined
:One of the two prime factors of the $8$th Fermat number $2^{\paren {2^8} } + 1$:
::$1 \, 238 \, 926 \, 361 \, 552 \, 897 = 1 \, 209 \, 889 \, 024 \, 954 \times 2^{10} + 1$
== Also see ==
* Prime Decomposition of 8th Fermat Number
== Sources ==
* : $257$
* : $257$
Category:Specific Numbers
Category:1,238,926,361,552,897
1,257,787

== Number ==
$1 \, 257 \, 787$ (one million, two hundred and fifty-seven thousand, seven hundred and eighty-seven) is:
:The $97 \, 017$th prime number
:The index of the $34$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $\ldots$, $23 \, 209$, $44 \, 497$, $86 \, 243$, $110 \, 503$, $132 \, 049$, $216 \, 091$, $756 \, 839$, $859 \, 433$:
::$M_{1 \, 257 \, 787} = 2^{1 \, 257 \, 787} - 1 \approx 4 \cdotp 122 \times 10^{378 \, 631}$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* : $2^{1,257,787} - 1$
Category:Specific Numbers
Category:1,257,787
1,257,787/Historical Note

== Historical Note on $1 \, 257 \, 787$ ==
The Mersenne number $M_{1 \, 257 \, 787} = 2^{1 \, 257 \, 787} - 1$ was demonstrated to be a Mersenne prime by and .
This discovery was announced on $2$ September $1996$.
== Sources ==
* : $2^{1,257,787} - 1$
* 
Category:Historical Notes
1,269,621

== Number ==
$1 \, 269 \, 621$ (one million, two hundred and sixty-nine thousand, six hundred and twenty-one) is:
:$3^3 \times 59 \times 797$
:The $17$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$, $666$, $3003$, $5995$, $8778$, $15 \, 051$, $66 \, 066$, $617 \, 716$, $828 \, 828$:
::$1 \, 269 \, 621 = \dfrac {1593 \times \paren {1593 + 1} } 2$
:The $1593$rd triangular number:
::$1 \, 269 \, 621 = \dfrac {1593 \times \paren {1593 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1,269,621
1,322,314,049,613,223,140,496

== Number ==
$1 \, 322 \, 314 \, 049 \, 613 \, 223 \, 140 \, 496$ is:
:$2^4 \times 11^2 \times 23^2 \times 4093^2 \times 8779^2$
The $36 \, 363 \, 636 \, 364$th square number:
::$1 \, 322 \, 314 \, 049 \, 613 \, 223 \, 140 \, 496 = 36 \, 363 \, 636 \, 364 \times 36 \, 363 \, 636 \, 364$
:A square number which can be split into two identical halves:
::$1 \, 322 \, 314 \, 049 \, 613 \, 223 \, 140 \, 496 = 36 \, 363 \, 636 \, 364^2$
== Also see ==
* Number whose Square is in 2 Identical Halves
* 
== Sources ==
* : $36,363,636,364$
* : $36,363,636,364$
Category:Specific Numbers
Category:1,322,314,049,613,223,140,496
1,367,631

== Number ==
$1 \, 367 \, 631$ (one million, three hundred and sixty-seven thousand, six hundred and thirty-one) is:
:$3^3 \times 37^3$
:The $111$th cube number:
::$1 \, 367 \, 631 = 111 \times 111 \times 111$
:The $6$th palindromic cube after $1$, $8$, $343$, $1331$, $1 \, 030 \, 301$:
::$1 \, 367 \, 631 = 111^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1,367,631
1,375,298,099

== Number ==
$1 \, 375 \, 298 \, 099$ is:
:$7 \times 196 \, 471 \, 157$
:The smallest positive integer which can be expressed as the sum of $3$ fifth powers in $2$ different ways:
::$1 \, 375 \, 298 \, 099 = 24^5 + 28^5 + 67^5 = 3^5 + 54^5 + 62^5$
== Also see ==
* Smallest Integer which is Sum of 3 Fifth Powers in 2 Ways
== Sources ==
* : $1,375,298,099$
* : $1,375,298,099$
Category:Specific Numbers
Category:1,375,298,099
1,398,269

== Number ==
$1 \, 398 \, 269$ (one million, three hundred and ninety-eight thousand, two hundred and sixty-nine) is:
:The $106 \, 991$st prime number
:The index of the $35$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $\ldots$, $23 \, 209$, $44 \, 497$, $86 \, 243$, $110 \, 503$, $132 \, 049$, $216 \, 091$, $756 \, 839$, $859 \, 433$, $1 \, 257 \, 787$:
::$M_{1 \, 398 \, 269} = 2^{1 \, 398 \, 269} - 1 \approx 8 \cdotp 147 \times 10^{420 \, 920}$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* : $2^{1,398,269} - 1$
Category:Specific Numbers
Category:1,398,269
1,398,269/Historical Note

== Historical Note on $1 \, 398 \, 269$ ==
The Mersenne number $M_{1 \, 398 \, 269} = 2^{1 \, 398 \, 269} - 1$ was demonstrated to be a Mersenne prime by , as part of the GIMPS team headed by .
This discovery was announced on $13$ November $1996$.
== Sources ==
* : $2^{1,398,269} - 1$
Category:Historical Notes
1,425,600

== Number ==
$1 \, 425 \, 600$ (one million, four hundred and twenty-five thousand, six hundred) is:
:$2^6 \times 3^4 \times 5^2 \times 11$
:The common product of the only known set of $5$ triplets of integers with the property that the sum of the integers in each triplet are equal and the product of the integers in each triplet are equal:
::$\tuple {6, 480, 495}$, $\tuple {11, 160, 810}$, $\tuple {12, 144, 825}$, $\tuple {20, 81, 880}$, $\tuple {33, 48, 900}$
== Also see ==
* Set of 5 Triplets whose Sums and Products are Equal
* 981
== Sources ==
* : $981$
* : $981$
Category:Specific Numbers
Category:1,425,600
1,438,983

== Number ==
$1 \, 438 \, 983$ (one million, four hundred and thirty-eight thousand, nine hundred and eighty-three) is:
:$3^2 \times 7^2 \times 13 \times 251$
:The smaller of the $9$th odd amicable pair, with $1 \, 438 \, 983$:
::$\map {\sigma_1} {1 \, 175 \, 265} = \map {\sigma_1} {1 \, 438 \, 983} = 2 \, 614 \, 248 = 1 \, 175 \, 265 + 1 \, 438 \, 983$
== Also see ==
* 
== Sources ==
* : $1,175,265$
* : $1,175,265$
Category:Specific Numbers
Category:1,438,983
1,476,304,896

== Number ==
$1 \, 476 \, 304 \, 896$ is:
:$2^{13} \times 3 \times 11 \times 43 \times 127$
:The $5$th triperfect number after $120$, $672$, $523 \, 776$, $459 \, 818 \, 240$:
::$\map {\sigma_1} {1 \, 476 \, 304 \, 896} = 4 \, 428 \, 914 \, 688 = 3 \times 1 \, 476 \, 304 \, 896$
== Also see ==
* 
Category:Specific Numbers
Category:1,476,304,896
1,480,028,171

== Number ==
$1 \, 480 \, 028 \, 171$ is:
:The central element of the smallest prime magic square whose elements are consecutive odd primes:
== Also see ==
* Smallest Order $3$ Prime Magic Square with Consecutive Primes
* 
== Sources ==
* : $1,480,028,171$
Category:Specific Numbers
Category:1,480,028,171
1,533,776,805

== Number ==
$1 \, 533 \, 776 \, 805$ is:
:$3^2 \times 5 \times 11 \times 17 \times 19 \times 53 \times 181$
:The $3$rd integer after $1$, $40 \, 755$ to be both pentagonal and hexagonal
:The $5$th integer after $1$, $210$, $40 \, 755$, $7 \, 906 \, 276$ to be both pentagonal and triangular
:The $27 \, 693$rd hexagonal number:
::$1 \, 533 \, 776 \, 805 = \ds \sum_{k \mathop = 1}^{27 \, 693} \paren {4 k - 3} = 27 \, 693 \paren {2 \times 27 \, 693 - 1}$
:The $31 \, 977$th pentagonal number:
::$1 \, 533 \, 776 \, 805 = \ds \sum_{k \mathop = 1}^{31 \, 977} \paren {3 k - 2} = \dfrac {31 \, 977 \paren {3 \times 31 \, 977 - 1} } 2$
:The $55 \, 385$th triangular number:
::$1 \, 533 \, 776 \, 805 = \ds \sum_{k \mathop = 1}^{55 \, 385} k = \dfrac {55 \, 385 \times \paren {55 \, 385 + 1} } 2$
:The $63 \, 953$rd generalized pentagonal number:
::$1 \, 533 \, 776 \, 805 = \ds \sum_{k \mathop = 1}^{31 \, 977} \paren {3 k - 2} = \dfrac {31 \, 977 \paren {3 \times 31 \, 977 - 1} } 2$
== Also see ==
* 
* 
* 
* 
* 
* 
== Sources ==
* : $1,533,776,805$
* : $1,533,776,805$
Category:Specific Numbers
Category:1,533,776,805
1,594,323

== Number ==
$1 \, 594 \, 323 $ (one million, five hundred and ninety-four thousand, three hundred and twenty-three) is:
:$3^{13}$
:The $13$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$, $243$, $729$, $2187$, $6561$, $19 \, 683$, $59 \, 049$, $177 \, 147$, $531 \, 441$:
::$1 \, 594 \, 323 = 3^{13}$
== Also see ==
* 
Category:Specific Numbers
Category:1,594,323
1,679,616

== Number ==
$1 \, 679 \, 616$ (one million, six hundred and seventy-nine thousand, six hundred and sixteen) is:
:$2^8 \times 3^8$
:The $1296$th square number:
::$1 \, 679 \, 616 = 1296 \times 1296$
:The $36$th fourth power:
::$1 \, 679 \, 616 = 36 \times 36 \times 36 \times 36$
:The $6$th eighth power after $1$, $256$, $6561$, $65 \, 536$, $390 \, 625$:
::$1 \, 679 \, 616 = 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6$
:The $8$th power of $6$ after after $(1)$, $6$, $36$, $216$, $1296$, $7776$, $46 \, 656$, $279 \, 936$:
::$1 \, 679 \, 616 = 6^8$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1,679,616
1,680,861

== Number ==
$1 \, 680 \, 861$ (one million, six hundred and eighty thousand, eight hundred and sixty-one) is:
:$3 \times 7 \times 13 \times 47 \times 131$
:The $18$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$, $666$, $3003$, $5995$, $8778$, $15 \, 051$, $66 \, 066$, $617 \, 716$, $828 \, 828$, $1 \, 269 \, 621$:
::$1 \, 680 \, 861 = \dfrac {1833 \times \paren {1833 + 1} } 2$
:The $1833$rd triangular number:
::$1 \, 680 \, 861 = \dfrac {1833 \times \paren {1833 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1,680,861
1,741,725

== Number ==
$1 \, 741 \, 725$ (one million, seven hundred and forty-one thousand, seven hundred and twenty-five) is:
:$3^2 \times 5^2 \times 7741$
:The $21$st pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $8208$, $9474$, $54 \, 748$, $92 \, 727$, $93 \, 084$, $548 \, 834$:
== Also see ==
* 
== Sources ==
* : $1,741,725$
* : $1,741,725$
Category:Specific Numbers
Category:1,741,725
1,747,515

== Number ==
$1 \, 747 \, 515$ (one million, seven hundred and forty-seven thousand, five hundred and fifteen) is:
:$3 \times 5 \times 7 \times 11 \times 17 \times 89$
:The $1$st of the $26$th pair of triangular numbers whose sum and difference are also both triangular:
::$1 \, 747 \, 515 = T_{1869}$, $2 \, 185 \, 095 = T_{2090}$, $2 \, 185 \, 095 + 1 \, 747 \, 515 = T_{2804}$, $2 \, 185 \, 095 - 1 \, 747 \, 515 = T_{935}$
:The $1869$th triangular number:
::$1 \, 747 \, 515 = \ds \sum_{k \mathop = 1}^{1869} k = \dfrac {1869 \times \paren {1869 + 1} } 2$
== Also see ==
* 
* 
== Sources ==
* : $1,747,515$
* : $1,747,515$
Category:Specific Numbers
Category:1,747,515
1,771,561

== Number ==
$1 \, 771 \, 561$ (one million, seven hundred and seventy-one thousand, five hundred and sixty-one) is:
:$11^6$
:The $4$th, and highest known, odd number after $1$, $5$, $1549$ which cannot be expressed as the sum of an integer power and a prime number
:The $6$th power of $11$ after $(1)$, $11$, $121$, $1331$, $14 \, 641$, $161 \, 051$:
::$1 \, 771 \, 561 = 11^6$
:The $11$th sixth power after $1$, $64$, $729$, $4096$, $15 \, 625$, $46 \, 656$, $117 \, 649$, $262 \, 144$, $531 \, 441$, $1 \, 000 \, 000$:
::$1 \, 771 \, 561 = 11 \times 11 \times 11 \times 11 \times 11 \times 11$
:The $121$st cube number:
::$1 \, 771 \, 561 = 121 \times 121 \times 121$
:The $1331$st square number:
::$1 \, 771 \, 561 = 1331 \times 1331$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:1,771,561
1,786,772,701,928,802,632,268,715,130,455,793

== Number ==
$1 \, 786 \, 772 \, 701 \, 928 \, 802 \, 632 \, 268 \, 715 \, 130 \, 455 \, 793$ is:
:$2521 \times 49 \, 993 \times 14 \, 177 \, 095 \, 479 \, 037 \, 851 \, 751 \, 198 \, 481$
:The $1$st term of a general Fibonacci sequence whose first $2$ terms are coprime such that all its terms were mistakenly believed at one time to be composite:
::$1 \, 786 \, 772 \, 701 \, 928 \, 802 \, 632 \, 268 \, 715 \, 130 \, 455 \, 793$, $1 \, 059 \, 683 \, 225 \, 053 \, 915 \, 111 \, 058 \, 165 \, 141 \, 686 \, 995$, $2 \, 846 \, 455 \, 926 \, 982 \, 717 \, 743 \, 326 \, 880 \, 272 \, 142 \, 788$, $\ldots$
== Also see ==
* 
== Sources ==
* : $1,786,772,701,928,802,632,268,715,130,455,793$
* : $1,786,772,701,928,802,632,268,715,130,455,793$
Category:Specific Numbers
Category:1,786,772,701,928,802,632,268,715,130,455,793
1,787,109,376

== Number ==
$1 \, 787 \, 109 \, 376$ is:
:$2^{13} \times 13 \times 97 \times 173$
:The $18$th automorphic number after $1$, $5$, $6$, $25$, $76$, $376$, $625$, $9376$, $\ldots$, $12 \, 890 \, 625$, $87 \, 109 \, 376$, $212 \, 890 \, 625$, $787 \, 109 \, 376$:
::$1 \, 787 \, 109 \, 376^2 = 3 \, 193 \, 759 \, 92 \, \mathbf {1 \, 787 \, 109 \, 376}$
:and the $1$st of the only $2$ such with $10$ digits.
== Also see ==
* 
== Sources ==
* : $1,787,109,376$
* : $1,787,109,376$
Category:Specific Numbers
Category:1,787,109,376
1,857,437,604

== Number ==
$1 \, 857 \, 437 \, 604$ is:
:$2^2 \times 3^2 \times 11^2 \times 653^2$
:The $43 \, 098$th square number:
::$1 \, 857 \, 437 \, 604 = 43 \, 098 \times 43 \, 098$
:A square number whose divisor sum is a cube:
::$\map {\sigma_1} {1 \, 857 \, 437 \, 604} = 5 \, 168 \, 743 \, 489 = 1729^3$
== Also see ==
* Square whose Divisor Sum is Cubic
* 
== Sources ==
* : $1,857,437,604$
* : $1,857,437,604$
Category:Specific Numbers
Category:1,857,437,604
Category:Integers whose Divisor Sum is Cube
1,979,339,333

== Number ==
$1 \, 979 \, 339 \, 333$ is:
:The largest but one right-truncatable prime number, if $1$ is temporarily considered to be a prime number:
::$1 \, 979 \, 339 \, 333$, $197 \, 933 \, 933$, $\ldots$, $19 \, 793$, $1979$, $197$, $19$ are all prime
== Also see ==
* 
* Largest Right-Truncatable Primes allowing 1
== Sources ==
* : $1,979,339,339$
* : $1,979,339,339$
Category:Specific Numbers
Category:1,979,339,333
1,979,339,339

== Number ==
$1 \, 979 \, 339 \, 339$ is:
:The largest right-truncatable prime number, if $1$ is temporarily considered to be a prime number:
::$1 \, 979 \, 339 \, 339$, $197 \, 933 \, 933$, $\ldots$, $19 \, 793$, $1979$, $197$, $19$ are all prime
== Also see ==
* 
* Largest Right-Truncatable Primes allowing 1
== Sources ==
* : $1,979,339,339$
* : $1,979,339,339$
Category:Specific Numbers
Category:1,979,339,339
1-Seminorm on Continuous on Closed Interval Real-Valued Functions is Norm

== Theorem ==
Let $C \closedint a b$ be the space of real-valued functions continuous on $\closedint a b$.
Let $x \in C \closedint a b$ be a continuous real valued function.
Let $\ds \norm x_1 := \int_a^b \size {\map x t} \rd t$ be the 1-seminorm.
Then $\norm {\, \cdot \,}_1$ is a norm on $C \closedint a b$.
== Proof ==
=== Positive definiteness ===
Let $x \in C \closedint a b$.
Then $\forall t \in \closedint 0 1 : \size {\map x t} \ge 0 $.
Hence:
:$\ds \int_a^b \size {\map x t} \rd t = \norm x_1 \ge 0$.
Suppose $\forall t \in \closedint a b : \map x t = 0$.
Then $\norm x_1 = 0$.
Therefore:
:$\paren {x = 0} \implies \paren {\norm x_1 = 0}$
Let $x \in C \closedint a b : \norm x_1 = 0$.
Suppose:
:$\forall t \in \openint a b : \map x t = 0$.
By assumption of continuity of $x$:
:$\forall t \in \closedint a b : \map x t = 0$.
:
:$\exists t_0 \in \openint a b : \map x {t_0} \ne 0$.
By assumption, $x$ is continuous at $t_0$.
:$\forall \epsilon \in \R_{> 0} : \exists \delta \in \R_{> 0} : \size {t - t_0} < \delta \implies \size {\map x t - \map x {t_0} } < \epsilon$
Furthermore:
:$\exists \delta \in \R_{> 0} : \paren {a < t_0 - \delta} \land \paren {t_0 + \delta < b}$
Let $\ds \epsilon = \frac {\size {\map x {t_0} } } 2$.
We have that:
Hence:
Hence, we reached a contradiction.
Therefore:
:$\paren {\norm x_1 = 0} \implies \paren {x = 0}$
=== Positive homogeneity ===
Let $x \in C \closedint a b$, $\alpha \in \R$.
Then:
=== Triangle inequality ===
Let $x, y \in C \closedint a b$
== Also see ==
* Definition:P-Seminorm
== Sources ==
* : $\S 1.2$: Normed and Banach spaces. Normed Spaces
Category:Examples of Norms
1-Sequence Space is Proper Subset of 2-Sequence Space

== Theorem ==
Let $\ell^1$ and $\ell^2$ be the $1$-sequence space and $2$-sequence space respectively.
Then $\ell^1$ is a proper subset of $\ell^2$.
== Proof ==
=== $\ell^1$ is a subset of $\ell^2$ ===
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $\ell^1$.
By definition of $1$-sequence space:
:$\ds \sum_{n \mathop = 0}^\infty \size {x_n} < \infty$
By Terms in Convergent Series Converge to Zero:
:$\ds \lim_{n \mathop \to \infty} \size {x_n} = 0$
By definition of convergent sequence:
:$\exists N \in \N : \forall n > N : \size {x_n} < 1$
Furthermore:
:$\forall \size {x_n} < 1 : \size {x_n}^2 < \size {x_n}$
By Comparison Test:
:$\ds \sum_{n \mathop = 0}^\infty \size {x_n}^2 < \infty$
Hence:
:$\sequence {x_n}_{n \mathop \in \N} \in \ell^2$
That is:
:$\ell^1 \subseteq \ell^2$
=== $\ell^2$ is not a subset of $\ell^1$ ===
Let $x := \tuple {1, \dfrac 1 2, \ldots, \dfrac 1 n, \ldots}$ with $n \in \N_{>0}$.
From Basel Problem:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^2} < \infty$
From Harmonic Series is Divergent:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 n$ 
diverges.
Hence:
:$x \in \ell^2$
but:
:$x \notin \ell^1$
That is:
:$\ell^2 \nsubseteq \ell^1$
By definition, $\ell^1$ and $\ell^2$ are not equal.
Therefore:
:$\ell^1 \subsetneqq \ell^2$
== Sources ==
* : $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces
Category:Proper Subsets
1-Sequence Space is Separable

== Theorem ==
$\ell^1$ space is a separable space.
== Proof ==
Let $D$ be the set of all finitely supported sequences with rational terms:
:$D = \set {\sequence {q_i}_{i \mathop \in \N} : n \in \N : i \le n : q_i \in \Q}$
We have that:
:Rational Numbers are Countably Infinite
:A finite set is countable
:$D$ is a union of finite sets indexed by $n$, which is countable
By Countable Union of Countable Sets is Countable, $D$ is countable.
Let $\mathbf x := \tuple {x_n}_{n \mathop \in \N} \in \ell^1$
By definition of $\ell^1$:
:$\ds \sum_{n \mathop = 1}^\infty \size {x_n} < \infty$
Let $\ds s_n := \sum_{i \mathop = 0}^n \size {x_i}$ be a sequence of partial sums of $\ds s = \sum_{i \mathop = 0}^\infty \size {x_i}$.
Then $s$ is a convergent sequence:
:$\forall \epsilon' \in \R_{>0}: \exists N \in \N: \forall n \in \N: n > N \implies \size {s_n - s} < \epsilon'$
Note that:
Let $\epsilon' = \dfrac \epsilon 2$.
Let $\ds N \in \N : \sum_{n \mathop = N + 1}^\infty \size {x_n} < \frac \epsilon 2$.
We have that $\Q$ is dense in $\R$.
:$\forall x \in \R : \exists \epsilon \in \R_{\mathop > 0} : \exists q \in \Q : \size {x - q} < \epsilon$
Then:
:$\ds \forall i \in \N : 1 \le i \le N : q_i \in \Q : \sum_{n \mathop = 1}^N \size {x_n - q_n} < \frac \epsilon 2$
where
:$\ds \frac \epsilon 2 = \sum_{i \mathop = 1}^N \epsilon_i$
Let $\mathbf x' := \tuple {q_1, \ldots, q_N, 0, \ldots} \in D$.
Then:
:$\ds \norm {\mathbf x - \mathbf x '} = \sum_{n \mathop = 1}^N \size {x_n - q_n} + \sum_{n \mathop = N + 1}^\infty \size {x_n} < \epsilon$
== Sources ==
* : Chapter $\S 1.3$: Normed and Banach spaces. Topology of normed spaces
Category:Separable Spaces
1/Historical Note

== Historical Note on $1$ (one) ==
The ancient Greeks did not consider $1$ to be a number.
According to the Pythagoreans, the number One ($1$) was the Generator of all Numbers: the omnipotent One.
It represented reason, for reason could generate only $1$ self-evident body of truth.
While a number, according to , was an aggregate of units, a unit was not considered to be an aggregate of itself.
The much-quoted statement of might as well be repeated here:
:Wherefrom thou understandest that $1$ is no number but it is a generatrix beginning and foundation for all other numbers.
:::: -- $1537$
illustrating that this mindset still held sway as late as the $16$th century.
The ancient Greeks considered $1$ as both odd and even by fallacious reasoning.
== Sources ==
* 
* : Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory
* : $1$
* : Chapter $\text {A}.2$: Pythagoras (ca. B.C.)
* : $1$
* : Chapter $2$: The Logic of Shape: Pythagoras
Category:Historical Notes
1/Linguistic Note

== Linguistic Note on $1$ (one) ==
Words derived from or associated with the number $1$ include:
:From English one, from Germanic
:*alone: literally: all one
:*only: literally: one-like
:*lonely: a combination of alone and only, hence all one-like
:*atone: to repent of one's misdeeds by becoming at one with the person you have offended against
:*none: literally not one
:*once: meaning one time
:*nonce, as in for the nonce, or nonce word: meaning just for this one time
:From Latin unum
:*united: many things that have become one
:*union: an arrangement where the members are unified into one body
:*unanimous: where people speak with one spirit
:*universal: as in universal principle, for example: holding throughout the universe
:*university: students and professors are turned into one body
:*uniform: clothes all of one form
:*unicorn: a mythical beast with one horn
:*onion: from the same root as union: the Romans referred to it as one large pearl
:From Greek μόνος (mónos)
:*monologue: a speech by $1$ person
:*monopoly: selling by only $1$ agency
:*monk: from the Greek monakos: someone who is alone or solitary
:*monolith: something made from one stone
:*monogram: a way to write your name with just one drawing of the pen
:*monotonous: all of one (boring) type
== Sources ==
* : Chapter $1$: The Romance of Numbers: Number Words
* : Chapter $1$: The Romance of Numbers: $1$
Category:Linguistic Notes
10,000

== Number ==
$10 \, 000$ (ten thousand) is:
:$2^4 \times 5^4$
:The $4$th power of $10$ after $(1)$, $10$, $100$, $1000$:
::$10 \, 000 = 10^4$
:The $10$th fourth power after $1$, $16$, $81$, $256$, $625$, $1296$, $2401$, $4096$, $6561$:
::$10 \, 000 = 10 \times 10 \times 10 \times 10$
:The $37$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $5184$, $5776$, $6084$, $6400$, $7056$, $7744$, $8100$, $9216$, $9604$ to be the divisor sum value of some (strictly) positive integer:
::$10 \, 000 = \map {\sigma_1} {8743} = \map {\sigma_1} {9481}$
:The $100$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $9025$, $9216$, $9409$, $9604$, $9801$:
::$10 \, 000 = 100 \times 100$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : Glossary
* : Glossary
Category:Specific Numbers
Category:10,000
10,000,000

== Number ==
$10 \, 000 \, 000$ (ten million) is:
:$2^7 \times 5^7$
:The $7$th power of $10$ after $(1)$, $10$, $100$, $1000$, $10 \, 000$, $100 \, 000$, $1 \, 000 \, 000$:
::$10 \, 000 \, 000 = 10^7$
:The $10$th seventh power after $1$, $64$, $729$, $4096$, $15 \, 625$, $46 \, 656$, $117 \, 649$, $262 \, 144$, $531 \, 441$, $4 \, 782 \, 969$:
::$10 \, 000 \, 000 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10$
== Also see ==
* 
* 
Category:Specific Numbers
Category:10,000,000
10,001

== Number ==
$10 \, 001$ (ten thousand and one) is:
:$73 \times 137$
:The $9$th positive integer after $1$, $2$, $7$, $11$, $101$, $111$, $1001$, $2201$ whose cube is palindromic:
::$10 \, 001^3 = 1 \, 000 \, 300 \, 030 \, 001$
:The $15$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $111$, $121$, $202$, $212$, $1001$, $1111$, $2002$ whose square is also palindromic integer
::$10 \, 001^2 = 100 \, 020 \, 001$
== Also see ==
* 
* 
* 
== Sources ==
* : $10,001$
* : $10,001$
Category:Specific Numbers
Category:10,001
10,101

== Number ==
$10 \, 101$ (ten thousand, one hundred and one) is:
:$3 \times 7 \times 13 \times 37$
:The $10$th positive integer after $1$, $2$, $7$, $11$, $101$, $111$, $1001$, $2201$, $10 \, 001$ whose cube is palindromic:
::$10 \, 101^3 = 1 \, 030 \, 607 \, 060 \, 301$
:The $16$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $111$, $121$, $202$, $212$, $1001$, $1111$, $2002$, $10 \, 001$ whose square is also palindromic integer
::$10 \, 101^2 = 102 \, 030 \, 201$
== Also see ==
* 
* 
Category:Specific Numbers
Category:10,101