avewright/proofwiki-math huggingface dataset:
Theorem: Convergent Sequence in Normed Vector Space is Bounded

Let $\GF \in \set {\R, \C}$. 
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. 
Let $\sequence {x_n}_{n \in \N}$ be a convergent sequence. 
Then there exists $M > 0$ such that: 
:$\norm {x_n} \le M$
for each $n \in \N$.

Proof:
Suppose that $x_n \to x$. 
From Convergent Sequence is Cauchy Sequence, $\sequence {x_n}_{n \in \N}$ is a Cauchy sequence.
So there exists $N \in \N$ such that:
:$\norm {x_n - x_m} < 1$ 
for each $n, m \ge N$. 
That is: 
:$\norm {x_n - x_N} < 1$
for $n \ge N$. 
From Reverse Triangle Inequality: Normed Vector Space, we have: 
:$\norm {x_n} - \norm {x_N} < 1$
so that: 
:$\norm {x_n} < 1 + \norm {x_N}$
for $n \ge N$. 
Now set: 
:$M = \max \set {\norm {x_1}, \norm {x_2}, \ldots, \norm {x_{N - 1} }, 1 + \norm {x_N} }$ 
Then we have: 
:$\norm {x_n} \le M$
for each $n \in \N$. 
{{qed}}
Category:Convergent Sequences (Normed Vector Spaces)
sn68f42fff3x6joezemo8j0hqrrmc9c
Theorem: Sum Rule for Sequence in Normed Vector Space

Let $\GF \in \set {\R, \C}$. 
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$. 
Let $\sequence {x_n}_{n \in \N}$ and $\sequence {y_n}_{n \in \N}$ be convergent sequences such that: 
:$x_n \to x$
and:
:$y_n \to y$
Then: 
:$x_n + y_n \to x + y$

Proof:
For each $n \in \N$, we have: 
{{begin-eqn}}
{{eqn	| l = \norm {\paren {x_n + y_n} - \paren {x + y} }
		| r = \norm {\paren {x_n - x} + \paren {y_n - y} }
}}
{{eqn	| o = \le
		| r = \norm {x_n - x} + \norm {y_n - y}
		| c = {{NormAxiomVector|3}}
}}
{{eqn	| o = \to
		| r = 0
}}
{{end-eqn}}
So from Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence, we have: 
:$x_n + y_n \to x + y$
{{qed}}
Category:Convergent Sequences (Normed Vector Spaces)
1opv7eim4e0tffkrdhufljiwm9afh7x
Theorem: Reciprocal of 13

:$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3 \ldots$

Proof:
Performing the calculation using long division:
<pre>
   0.07692307...
  -------------
13)1.00000000...
     91            
   ----    
      90
      78
      --
      120
      117
      ---
        30
        26
        --
         40
         39
         --
          100
           91
          ---
          ...
</pre>
{{qed}}
Category:13
Category:Examples of Reciprocals
5ha7983p4szwc6onmklyak0k4lrn71q
Theorem: Reciprocal of 17

:$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$

Proof:
Performing the calculation using long division:
<pre>
   0.058823529411764705...
  ------------------------
17)1.000000000000000000000
     85      68
     --      --    
     150      20        
     136      17
     ---      --
      140      30
      136      17
      ---      --
        40     130
        34     119
        --     ---
         60     110
         51     102
         --     ---
          90      80
          85      68
          --      --
           50     120
           34     119
           --     ---
           160      100
           153       85
           ---      ---
             70     ...
             68
</pre>
{{qed}}
Category:17
Category:Examples of Reciprocals
4j0cnr55znegyw93ozsuupdij3isinr
Theorem: Reciprocal of 27

:$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$

Proof:
Performing the calculation using long division:
<pre>
   0.037...
  --------
27)1.00000
     81
     --
     190
     189
     ---
       100
        81
       ---
        ...
</pre>
It is to be noted that this is because $999 = 27 \times 37$.
{{qed}}
Category:27
Category:Examples of Reciprocals
dqrjepi3jc57oo9qvkiy5drw5iyvxuk
Theorem: Reciprocal of 101

:$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$

Proof:
Performing the calculation using long division:
<pre>
    0.00990099...
   --------------
101)1.00000000...
      909
      ---
       910
       909
       ---
         1000
          909
         ----
           910
           909
           ---
           ...
</pre>
{{qed}}
Category:101
Category:Examples of Reciprocals
rm5b2ayxmlia3a53vljeqiys4krqjob
Theorem: Reciprocal of 19

:$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$

Proof:
Performing the calculation using long division:
<pre>
   0.05263157894736842105...
  ------------------------
19)1.00000000000000000000000
     95    152     76 
     --    ---     --
      50    180     40
      38    171     38
      --    ---     --
      120     90     20
      114     76     19
      ---     --     --
        60    140     100
        57    133      95
        --    ---     ---
         30     70    ...
         19     57
         --     --
         110    130
          95    114
          --    ---
          150    160
          133    152
           --    ---
           170     80
           152     76
</pre>
{{qed}}
Category:19
Category:Examples of Reciprocals
algo7hfs63wonvl12payyuu8ht44oqs
Theorem: Reciprocal of 103

:$\dfrac 1 {103} = 0 \cdotp \dot 00970 \, 87378 \, 64077 \, 66990 \, 29126 \, 21359 \, 223 \dot 3$

Proof:
Performing the calculation using long division:
<pre>
    0.0097087378640776699029126213592233009...
  --------------------------------------------
103)1.0000000000000000000000000000000000000...
      927     618     927     103    309
      ---     ---     ---     ---    ---
       730     420      300    370     1000
       721     412      206    309      927
       ---     ---      ---    ---     ----
         900     800     940    610    ....
         824     721     927    515
         ---     ---     ---    ---
          760     790     130    950
          721     721     103    927
          ---     ---     ---    ---
           390     690     270    230
           309     618     206    206
           ---     ---     ---    ---
            810     720     640    240
            721     618     618    206
            ---     ---     ---    ---
             890    1020     220    340
             824     927     206    309
             ---    ----     ---    ---
              660     930     140    310
              618     927     103    309
</pre>
{{qed}}
Category:103
Category:Examples of Reciprocals
86c1eyisex9mm6npz9ucuyqyk135bey
Theorem: Reciprocal of 23

:$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$

Proof:
Performing the calculation using long division:
<pre>
   0.043478260869565217391304...
  -----------------------------
23)1.00000000000000000000000000
     92     184     23
     --     ---     --
      80     160    170
      69     138    161
      --     ---    ---
      110     220     90
       92     207     69
      ---     ---     --
       180     130    210
       161     115    207
       ---     ---    ---
        190     150     30
        184     138     23
        ---     ---     --
          60     120     70
          46     115     69
          --     ---     --
          140      50     100
          138      46      92
          ---      --     ---
            200     40    ...
            184     23
</pre>
{{qed}}
Category:23
Category:Examples of Reciprocals
kaew00no6a6cup57h3m05kpaqfspmj7
Theorem: Reciprocal of 29

:$\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$

Proof:
Performing the calculation using long division:
<pre>
   0.034482758620689655172413793103...
  ------------------------------------
29)1.000000000000000000000000000000...
     87
     -- 
     130    180     150    230
     116    174     145    203
     ---    ---     ---    ---
      140     60      50    270
      116     58      29    261
      ---     --      --    ---
       240     200    210     90
       232     174    203     87
       ---     ---    ---     --
         80     260     70     30
         58     232     58     29
         --     ---     --     --
         220     280    120     100
         203     261    116      87
          --     ---    ---     ---
          170     190     40    ...
          145     174     29
          ---     ---     --
           250     160    110
           232     145     87
           ---     ---    ---
            180     150    230
</pre>
{{qed}}
Category:29
Category:Examples of Reciprocals
rtaxpxd1ppw0hl932jotyqcamtiwv4e
Theorem: Reciprocal of 31

:$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$

Proof:
Performing the calculation using long division:
<pre>
   0.03225806451612903...
  -----------------------
31)1.00000000000000000000
     93     155
     --     ---
      70      50
      62      31
      --      --
       80     190
       62     186
       --     ---
       180      40
       155      31
       ---      --
        250      90
        248      62
        ---      --
          200    280
          186    279
          ---    ---
           140     100
           124      93
           ---     ---
            160    ...
            155
            ---
</pre>
{{qed}}
Category:31
Category:Examples of Reciprocals
qszev5doxzd900153j47qv48o4h3r6x
Theorem: Reciprocal of 37

:$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

Proof:
Performing the calculation using long division:
<pre>
   0.027...
  --------
37)1.00000
     74
     --
     260
     259
     ---
       100
        74
       ---
        ...
</pre>
It is to be noted that this is because $999 = 27 \times 37$.
{{qed}}
Category:37
Category:Examples of Reciprocals
b8x8urhl335q0n7odqvo8rn09s98sta
Theorem: Reciprocal of 41

:$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$

Proof:
Performing the calculation using long division:
<pre>
   0.0243902...
  -------------
41)1.0000000...
     82           
   ----   
     180
     164
     ---
      160
      123
      ---
       370
       369
       ---
         100
          82
         ---
         ...
</pre>
{{qed}}
Category:41
Category:Examples of Reciprocals
jodkvgy8isk8ull1p2x1twpk84eoumt
Theorem: Composition of Mapping with Mapping Restricted to Image

Let $A, B, C$ be sets.
Let $f : A \to B, g: B \to C$ be mappings.
Let $g \circ f : A \to C$ denote the composite mapping of $f$ with $g$.
Let $f \sqbrk A$ denote the image of $A$ under $f$.
Let $f \restriction_{A \times f \sqbrk A} : A \to f \sqbrk A$ denote the restriction of $f$ to $A \times f \sqbrk A$.
Let $g \restriction_{f \sqbrk A} : f \sqbrk A \to C$ denote the restriction of $g$ to $f \sqbrk A$.
Let $g \restriction_{f \sqbrk A} \circ f \restriction_{A \mathop \times f \sqbrk A} \mathop : A \to C$ denote the composite mapping of $f \restriction_{A \times f \sqbrk A}$ with $g \restriction_{f \sqbrk A}$.
Then:
:$g \restriction_{f \sqbrk A} \circ f \restriction_{A \mathop \times f \sqbrk A} \mathop = g \circ f$

Proof:
From Restriction of Mapping is Mapping:
:$f \restriction_{A \times f \sqbrk A} : A \to f \sqbrk A$ is a mapping
and
:$g \restriction_{f \sqbrk A} : f \sqbrk A \to C$ is a mapping
By definition of composite mapping:
:$g \restriction_{f \sqbrk A} \circ f \restriction_{A \mathop \times f \sqbrk A} \mathop : A \to C$  is a well-defined mapping
We have:
{{begin-eqn}}
{{eqn | q = \forall a \in A
      | l = \map {\paren{g \restriction_{f \sqbrk A} \circ f \restriction_{A \mathop \times f \sqbrk A} } } a
      | r = \map {g \restriction_{f \sqbrk A} } {\map {f \restriction_{A \mathop \times f \sqbrk A} } a }
      | c = {{Defof|Composite Mapping}}
}}
{{eqn | r = \map g {\map {f \restriction_{A \mathop \times f \sqbrk A} } a }
      | c = {{Defof|Restriction of Mapping}}
}}
{{eqn | r = \map g {\map f a }
      | c = {{Defof|Restriction of Mapping}}
}}
{{eqn | r = \map {\paren{g \circ f} } a
      | c = {{Defof|Composite Mapping}}
}}
{{end-eqn}}
From Equality of Mappings:
:$g \restriction_{f \sqbrk A} \circ f \restriction_{A \mathop \times f \sqbrk A} \mathop = g \circ f$
{{qed}}
Category:Mapping Theory
Category:Composite Mappings
Category:Restrictions
nw6nkzqlsm2jpdp0p85t3zba6o0osn3
Theorem: Reciprocal of 43

:$\dfrac 1 {43} = 0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \, \dot 3$

Proof:
Performing the calculation using long division:
<pre>
   0.02325581395348837209302...
  -----------------------------
43)1.00000000000000000000000...
     86        
   ----
     140    410    310
     129    387    301
     ---    ---    ---
      110    230     90
       86    215     86
      ---    ---     --
       240    150     400
       215    129     387
       ---    ---     ---
        250    210     130 
        215    172     129
        ---    ---     ---
         350    380      100
         344    344       86
         ---    ---      ---
           60    360     ...
           43    344
           --    ---
           170    160
           129    129
           ---    ---
            410    310
</pre>
{{qed}}
Category:43
Category:Examples of Reciprocals
325f1xlw4370bpccnd9ktnnmulorwxa
Theorem: Reciprocal of 53

:$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$

Proof:
Performing the calculation using long division:
<pre>
   0.018867924528301...
  ---------------------
53)1.000000000000000000
     53    212          
     --    ---
     470    280
     424    265
     ---    ---
      460    150
      424    106
      ---    ---
       360    440
       318    424
       ---    ---
        420    160
        371    159
        ---    ---
         490     100
         477      53
         ---     ---
          130    ...
          106
          ---
           240
           212
           ---
</pre>
{{qed}}
Category:53
Category:Examples of Reciprocals
6tz15rx7ltvgg353oodzh5d30wj2o8p
Theorem: Reciprocal of 49

:$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$

Proof:
Performing the calculation using long division:
<pre>
   0.02040816326530612244897959183673469387755102...
  ---------------------------------------------
49)1.00000000000000000000000000000000000000000000000
     98      294     196    441    196    245 
     --      ---     ---    ---    ---    ---
      200     260     240     90    340    250
      196     245     196     49    294    245
      ---     ---     ---     --    ---    ---
        400    150     440    410    460     50
        392    147     392    392    441     49
        ---    ---     ---    ---    ---     --
          80     300    480    180    190     100
          49     294    441    147    147      98
          --     ---    ---    ---    ---     ---
          310      60    390    330    430    ...
          294      49    343    294    392
          ---      --    ---    ---    ---
           160     110    470    360    380
           147      98    441    343    343
           ---     ---    ---    ---    ---
            130     120    290    170    360
             98      98    245    147    343
             ---    ---    ---    ---    ---
             320     220    450    230    270
             294     196    441    196    245
             ---     ---    ---    ---    ---
</pre>
{{qed}}
Category:49
Category:Examples of Reciprocals
l3yyf4g8hrsdt116c2en4dzxdn543zd
Theorem: Long Period Prime/Examples/61

The prime number $61$ is a long period prime:
:$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$

Proof:
From Reciprocal of $61$:
{{:Reciprocal of 61}}
Counting the digits, it is seen that this has a period of recurrence of $60$.
Inspecting the expansion and counting the digits, we find that each one appears exactly $6$ times.
Hence the result.
{{qed}}
Theorem: Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1

Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a normed vector space over $\Bbb F$.
Let $N$ be a closed linear subspace of $X$. 
Let $\struct {X/N, \norm {\, \cdot \,}_{X/N} }$ be the normed quotient vector space associated with quotient vector space $X/N$.
Let $\pi : X \to X/N$ be the quotient mapping associated with $X/N$. 
Then:
:$\norm \pi_{\map B {X, X/N} } = 1$ 
where $\norm {\, \cdot \,}_{\map B {X, X/N} }$ denotes the norm on the space of bounded linear transformations.

Proof:
Let $B_X$ be the unit open ball in $\struct {X, \norm {\, \cdot \,} }$.
Let $B_{X/N}$ be the unit open ball in $\struct {X/N, \norm {\, \cdot \,} }$.
From Quotient Mapping is Bounded in Normed Quotient Vector Space, $\pi$ is bounded.
We have: 
{{begin-eqn}}
{{eqn	| l = \norm \pi_{\map B {X, X/N} }
		| r = \sup_{x \in B_X} \norm {\map \pi x}_{X/N}
}}
{{eqn	| r = \sup_{q \in \map \pi {B_X} } \norm q_{X/N} 
}}
{{eqn	| r = \sup_{q \in B_{X/N} } \norm q_{X/N}
		| c = Quotient Mapping Maps Unit Open Ball in Normed Vector Space to Unit Open Ball in Normed Quotient Vector Space
}}
{{eqn	| r = 1
}}
{{end-eqn}}
{{qed}}
Category:Quotient Mappings
Category:Normed Quotient Vector Spaces
6bu9nipqv6530jmzvglqhfibs81run6
Theorem: Reciprocal of 239

:$\dfrac 1 {239} = 0 \cdotp \dot 00418 \, 4 \dot 1$

Proof:
Performing the calculation using long division:
<pre>
    0.0041841004...
   -------------
239)1.0000000000...
      956           
     ----   
       440
       239
       ---
       2010
       1912
       ----
         980
         956
         ---
          240
          239
          ---
            1000
             956
            ----
             ...
</pre>
{{qed}}
Category:239
Category:Examples of Reciprocals
f0i0esagnd39c13z58zyq4w97asbkk3
Theorem: Reciprocal of 73

:$\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$

Proof:
Performing the calculation using long division:
<pre>
   0.0136986301...
  -------------
73)1.0000000000...
     73
   ----
     270
     219
     ---
      510
      438
      ---
       720
       657
       ---
        630
        584
        ---
         460
         438
         ---
          220
          219
          ---
            100
             73
            ---
            ...
</pre>
{{qed}}
Category:73
Category:Examples of Reciprocals
o4b3yi27bdp2dr88egss9utyg04pijg
Theorem: Reciprocal of 79

:$\dfrac 1 {79} = 0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$

Proof:
Performing the calculation using long division:
<pre>
   0.012658227848101...
  ---------------------
79)1.000000000000000...
     79
   ----
     210    670
     158    632
     ---    ---
      520    380
      474    316
      ---    ---
       460    640
       395    632
       ---    ---
        650     80
        632     79
        ---     --
         180     100
         158      79
         ---     ---
          220    ...
          158
          ---
           620
           553
           ---
            670
</pre>
{{qed}}
Category:79
Category:Examples of Reciprocals
d47uko9ydwfo0hbuzhct6v0ilfbxd8m
Theorem: Reciprocal of 59

:$\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$

Proof:
Performing the calculation using long division:
<pre>
   0.016949152542372881355932203389830508474576271186440677966101...
  ------------------------------------------------------------------
59)1.000000000000000000000000000000000000000000000000000000000000...
     59
   ----
     410   150   170   350    230     500   370   380    570
     354   118   118   295    177     472   354   354    531
     ---   ---   ---   ---    ---     ---   ---   ---    ---
      560   320   520   550    530     280   160   260    390
      531   295   472   531    472     236   118   236    354
      ---   ---   ---   ---    ---     ---   ---   ---    ---
       290   250   480   190    580     440   420   240    360
       236   236   472   177    531     413   413   236    354
       ---   ---   ---   ---    ---     ---   ---   ---    ---
        540   140    80   130    490     270    70    400    60
        531   118    59   118    472     236    59    354    59
        ---   ---    --   ---    ---     ---    --    ---    --
          90   220   210   120    180     340   110    460    100
          59   177   177   118    177     295    59    413     59
          --   ---   ---   ---    ---     ---   ---    ---    ---
          310   430   330    200    300    450   510    470   ...
          295   413   295    177    295    413   472    413
          ---   ---   ---    ---    ---    ---   ---    ---
           150   170   350    230     500   370   380    570
</pre>
{{qed}}
Category:59
Category:Examples of Reciprocals
akbe3bkqqhrtbsmu0xquc7htec8nly5
Theorem: Reciprocal of 67

:$\dfrac 1 {67} = 0 \cdotp \dot 01492 \, 53731 \, 34328 \, 35820 \, 89552 \, 23880 \, 59 \dot 7$

Proof:
Performing the calculation using long division:
<pre>
   0.01492537313432835820895522388059701...
  -----------------------------------------
67)1.00000000000000000000000000000000000...
     67
   ----
     330   210   560    640   590
     268   201   536    603   536
     ---   ---   ---    ---   ---
      620    90   240    370   540
      603    67   210    335   536
      ---    --   ---    ---   ---
       170   230   390    350    400
       134   201   335    335    335
       ---   ---   ---    ---    ---
        360   290   550    150    650
        335   268   536    134    603
        ---   ---   ---    ---    ---
         250   220   140    160    470
         201   201   134    134    469
         ---   ---   ---    ---    ---
          490   190    600   260     100
          469   134    536   201      67
          ---   ---    ---   ---     ---
           210   560    640   590    ...
</pre>
{{qed}}
Category:67
Category:Examples of Reciprocals
99efk47kiysok97gibr367oozfbxe52
Theorem: Reciprocal of 71

:$\dfrac 1 {71} = 0 \cdotp \dot 01408 \, 45070 \, 42253 \, 52112 \, 67605 \, 63380 \, 2816 \dot 9$

Proof:
Performing the calculation using long division:
<pre>
   0.0140845070422535211267605633802816901...
  -------------------------------------------
71)1.0000000000000000000000000000000000000...
     71
   ----
     290      160    80    400    580
     284      142    71    355    568
     ---      ---    --    ---    ---
       600     180    90    450    120
       568     142    71    426     71
       ---     ---    --    ---    ---
        320     380   190    240    490
        284     355   142    213    426
        ---     ---   ---    ---    ---
         360     250   480    270    640
         355     213   426    213    639
         ---     ---   ---    ---    ---
           500    370   540    570     100
           497    355   497    568      71
           ---    ---   ---    ---     ---
             300   150   430     200   ...
             284   142   426     142
             ---   ---   ---     ---
              160    80    400    580
</pre>
{{qed}}
Category:71
Category:Examples of Reciprocals
g2lnnjpjmu25fl3px3ej8or0gscx4p9
Theorem: Reciprocal of 83

:$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$

Proof:
Performing the calculation using long division:
<pre>
   0.0120481927710843373493975903614457831325301...
  -------------------------------------------------
83)1.0000000000000000000000000000000000000000000...
     83
   ----
     170    640    310   810    120   260   100
     166    581    249   747     83   249    83
     ---    ---    ---   ---    ---   ---   ---
       400   590    610   630    370   110 
       332   581    581   581    332    83
       ---   ---    ---   ---    ---   ---
        680    90    290   490    380   270
        664    83    249   415    332   249
        ---    --    ---   ---    ---   ---
         160    700   410   750    480   210
          83    664   332   747    415   166
         ---    ---   ---   ---    ---   ---
          770    360   780    300   650   440
          747    332   747    249   581   415
          ---    ---   ---    ---   ---   ---
           230    280   330    510   690   250
           166    249   249    498   664   249
           ---    ---   ---    ---   ---   ---
            640    310   810    120   260    100
</pre>
{{qed}}
Category:83
Category:Examples of Reciprocals
f1kwmzfrbt3u5qsiey1kifxw7xql2jc
Theorem: Long Period Prime/Examples/59

The prime number $59$ is a long period prime:
:$\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$

Proof:
From Reciprocal of $59$:
{{:Reciprocal of 59}}
Counting the digits, it is seen that this has a period of recurrence of $58$.
Hence the result.
{{qed}}
Category:59
Category:Examples of Long Period Primes
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