# MathML formulae – Schlage-Puchta, Jan-Christoph: Finiteness of a class of Rabinowitsch polynomials. (English). Archivum Mathematicum, vol. 40 (2004), issue 3, pp. 259-261

## Formula 1 ¶

### Original ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Canonicalized ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + m - m + m - m + m m [/itex] 

## Formula 2 ¶

### Original ¶

$t$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  t t t t [/itex] 

### Canonicalized ¶

$t$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  t t t t [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + t - t + t - t + t t [/itex] 

## Formula 3 ¶

### Original ¶

$|n^{2}+n-m|$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  | n 2 + n - m | superscript n 2 n m - + n 2 n m |n^{2}+n-m| [/itex] 

### Canonicalized ¶

$|n^{2}+n-m|$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  | n 2 + n - m | superscript n 2 n m - + n 2 n m |n^{2}+n-m| [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 @@ -1,58 +1,58 @@ - - + + - | + | n 2 + n - m - | + | - superscript - n - 2 + superscript + n + 2 - n + n - m + m - + - - + - - + + + - - n - 2 + + n + 2 - n + n - m + m |n^{2}+n-m| [/itex] 

## Formula 4 ¶

### Original ¶

$n\in[t+1,t+\sqrt{m}]$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75  n [ t + 1 , t + m ] n t 1 t m n + t 1 + t m n\in[t+1,t+\sqrt{m}] [/itex] 

### Canonicalized ¶

$n\in[t+1,t+\sqrt{m}]$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75  n [ t + 1 , t + m ] n t 1 t m n + t 1 + t m n\in[t+1,t+\sqrt{m}] [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 @@ -1,75 +1,75 @@ - - + + n [ t + 1 , t + m ] - n + n - t - 1 + t + 1 - t + t - m + m - - n + + n - + - + - t - 1 + + + t + 1 - + - t + + + t - - m + + m n\in[t+1,t+\sqrt{m}] [/itex] 

## Formula 5 ¶

### Original ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Canonicalized ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + m - m + m - m + m m [/itex] 

## Formula 6 ¶

### Original ¶

$4m-1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  4 m - 1 4 m 1 - 4 m 1 4m-1 [/itex] 

### Canonicalized ¶

$4m-1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  4 m - 1 4 m 1 - 4 m 1 4m-1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 @@ -1,39 +1,38 @@ - - + + 4 - m - 1 - 4 - m + 4 + m - 1 + 1 - - + - - - 4 - m + + 4 + m - 1 + 1 4m-1 [/itex] 

## Formula 7 ¶

### Original ¶

$\mathbb{Q}(\sqrt{1-4m})$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63  ( 1 - 4 m ) 1 4 m Q - 1 4 m \mathbb{Q}(\sqrt{1-4m}) [/itex] 

### Canonicalized ¶

$\mathbb{Q}(\sqrt{1-4m})$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59  Q ( 1 - 4 m ) Q 1 4 m Q - 1 4 m \mathbb{Q}(\sqrt{1-4m}) [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 @@ -1,63 +1,59 @@ - - + + - - + Q ( + 1 + - - 1 - - - - 4 - - m - + 4 + m ) - + Q - 1 + 1 - 4 - m + 4 + m - - Q - - + + Q + + - - - 1 + - + 1 - - 4 - m + + 4 + m \mathbb{Q}(\sqrt{1-4m}) [/itex] 

## Formula 8 ¶

### Original ¶

$n^{2}+n+m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42  n 2 + n + m superscript n 2 n m + n 2 n m n^{2}+n+m [/itex] 

### Canonicalized ¶

$n^{2}+n+m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42  n 2 + n + m superscript n 2 n m + n 2 n m n^{2}+n+m [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 @@ -1,42 +1,42 @@ - - + + n 2 + n + m - superscript - n - 2 + superscript + n + 2 - n - m + n + m - + + + - - n - 2 + + n + 2 - n - m + n + m n^{2}+n+m [/itex] 

## Formula 9 ¶

### Original ¶

$0\leq n\leq m-3$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52  0 n m - 3 0 n m 3 0 n - m 3 0\leq n\leq m-3 [/itex] 

### Canonicalized ¶

$0\leq n\leq m-3$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52  0 n m - 3 0 n m 3 0 n - m 3 0\leq n\leq m-3 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 @@ -1,52 +1,52 @@ - - + + 0 n m - 3 - 0 - n + 0 + n - - - - m - 3 + + + + m + 3 - - 0 - - n - - - - - m - 3 + + 0 + + n + + + - + m + 3 0\leq n\leq m-3 [/itex] 

## Formula 10 ¶

### Original ¶

$f_{m}(x)=x^{2}+x-m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87  f m ( x ) = x 2 + x - m subscript f m x superscript x 2 x m = f m x - + x 2 x m f_{m}(x)=x^{2}+x-m [/itex] 

### Canonicalized ¶

$f_{m}(x)=x^{2}+x-m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86  f m ( x ) = x 2 + x - m subscript f m x superscript x 2 x m = f m x - + x 2 x m f_{m}(x)=x^{2}+x-m [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 @@ -1,87 +1,86 @@ - - + + f m - ( x ) = x 2 + x - m - subscript - f - m + subscript + f + m - x + x - superscript - x - 2 + superscript + x + 2 - x + x - m + m - = + = - + - - f - m + + f + m - x + x - - + - - + + + - - x - 2 + + x + 2 - x + x - m + m f_{m}(x)=x^{2}+x-m [/itex] 

## Formula 11 ¶

### Original ¶

$t$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  t t t t [/itex] 

### Canonicalized ¶

$t$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  t t t t [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + t - t + t - t + t t [/itex] 

## Formula 12 ¶

### Original ¶

$|f_{m}(n)|$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52  | f m ( n ) | subscript f m n f m n |f_{m}(n)| [/itex] 

### Canonicalized ¶

$|f_{m}(n)|$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51  | f m ( n ) | subscript f m n f m n |f_{m}(n)| [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 @@ -1,52 +1,51 @@ - - + + - | + | f m - ( n ) - | + | - subscript - f - m + subscript + f + m - n + n - + - + - - f - m + + f + m - n + n |f_{m}(n)| [/itex] 

## Formula 13 ¶

### Original ¶

$n\in[t+1,t+\sqrt{m}]$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75  n [ t + 1 , t + m ] n t 1 t m n + t 1 + t m n\in[t+1,t+\sqrt{m}] [/itex] 

### Canonicalized ¶

$n\in[t+1,t+\sqrt{m}]$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75  n [ t + 1 , t + m ] n t 1 t m n + t 1 + t m n\in[t+1,t+\sqrt{m}] [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 @@ -1,75 +1,75 @@ - - + + n [ t + 1 , t + m ] - n + n - t - 1 + t + 1 - t + t - m + m - - n + + n - + - + - t - 1 + + + t + 1 - + - t + + + t - - m + + m n\in[t+1,t+\sqrt{m}] [/itex] 

## Formula 14 ¶

### Original ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Canonicalized ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,26 +1,26 @@ - - + + f m - subscript - f - m + subscript + f + m - - f - m + + f + m f_{m} [/itex] 

## Formula 15 ¶

### Original ¶

$m=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  m = 1 m 1 = m 1 m=1 [/itex] 

### Canonicalized ¶

$m=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  m = 1 m 1 = m 1 m=1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,27 +1,27 @@ - - + + m = 1 - m - 1 + m + 1 - = - m - 1 + = + m + 1 m=1 [/itex] 

## Formula 16 ¶

### Original ¶

$m=2$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  m = 2 m 2 = m 2 m=2 [/itex] 

### Canonicalized ¶

$m=2$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  m = 2 m 2 = m 2 m=2 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,27 +1,27 @@ - - + + m = 2 - m - 2 + m + 2 - = - m - 2 + = + m + 2 m=2 [/itex] 

## Formula 17 ¶

### Original ¶

$m=p^{2}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  m = p 2 m superscript p 2 = m p 2 m=p^{2} [/itex] 

### Canonicalized ¶

$m=p^{2}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  m = p 2 m superscript p 2 = m p 2 m=p^{2} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 @@ -1,38 +1,38 @@ - - + + m = p 2 - m + m - superscript - p - 2 + superscript + p + 2 - = - m + = + m - - p - 2 + + p + 2 m=p^{2} [/itex] 

## Formula 18 ¶

### Original ¶

$p$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  p p p p [/itex] 

### Canonicalized ¶

$p$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  p p p p [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + p - p + p - p + p p [/itex] 

## Formula 19 ¶

### Original ¶

$m=t^{2}+t\pm 1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  m = t 2 + t ± 1 m plus-or-minus superscript t 2 t 1 = m ± + t 2 t 1 m=t^{2}+t\pm 1 [/itex] 

### Canonicalized ¶

$m=t^{2}+t\pm 1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  m = t 2 + t ± 1 m plus-or-minus superscript t 2 t 1 = m ± + t 2 t 1 m=t^{2}+t\pm 1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 @@ -1,60 +1,60 @@ - - + + m = t 2 + t ± 1 - m + m - plus-or-minus + plus-or-minus - superscript - t - 2 + superscript + t + 2 - t + t - 1 + 1 - = - m + = + m - ± + ± - + + + - - t - 2 + + t + 2 - t + t - 1 + 1 m=t^{2}+t\pm 1 [/itex] 

## Formula 20 ¶

### Original ¶

$m=t^{2}+t\pm\frac{2t+1}{3}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95  m = t 2 + t ± 2 t + 1 3 m plus-or-minus superscript t 2 t 2 t 1 3 = m ± + t 2 t + 2 t 1 3 m=t^{2}+t\pm\frac{2t+1}{3} [/itex] 

### Canonicalized ¶

$m=t^{2}+t\pm\frac{2t+1}{3}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94  m = t 2 + t ± 2 t + 1 3 m plus-or-minus superscript t 2 t 2 t 1 3 = m ± + t 2 t + 2 t 1 3 m=t^{2}+t\pm\frac{2t+1}{3} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 @@ -1,95 +1,94 @@ - - + + m = t 2 + t ± 2 - t + 1 3 - m + m - plus-or-minus + plus-or-minus - superscript - t - 2 + superscript + t + 2 - t + t - 2 - t + 2 + t - 1 + 1 - 3 + 3 - = - m + = + m - ± + ± - + + + - - t - 2 + + t + 2 - t + t - + - + + + - - 2 - t + + 2 + t - 1 + 1 - 3 + 3 m=t^{2}+t\pm\frac{2t+1}{3} [/itex] 

## Formula 21 ¶

### Original ¶

$\frac{2t+1}{3}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50  2 t + 1 3 2 t 1 3 + 2 t 1 3 \frac{2t+1}{3} [/itex] 

### Canonicalized ¶

$\frac{2t+1}{3}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49  2 t + 1 3 2 t 1 3 + 2 t 1 3 \frac{2t+1}{3} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 @@ -1,50 +1,49 @@ - - + + 2 - t + 1 3 - 2 - t + 2 + t - 1 + 1 - 3 + 3 - + - + + + - - 2 - t + + 2 + t - 1 + 1 - 3 + 3 \frac{2t+1}{3} [/itex] 

## Formula 22 ¶

### Original ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Canonicalized ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,26 +1,26 @@ - - + + f m - subscript - f - m + subscript + f + m - - f - m + + f + m f_{m} [/itex] 

## Formula 23 ¶

### Original ¶

$\mathbb{Q}(\sqrt{4m+1})$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63  ( 4 m + 1 ) 4 m 1 Q + 4 m 1 \mathbb{Q}(\sqrt{4m+1}) [/itex] 

### Canonicalized ¶

$\mathbb{Q}(\sqrt{4m+1})$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59  Q ( 4 m + 1 ) Q 4 m 1 Q + 4 m 1 \mathbb{Q}(\sqrt{4m+1}) [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 @@ -1,63 +1,59 @@ - - + + - - + Q ( - - 4 - - m - - + - 1 + 4 + m + + + 1 ) - + Q - 4 - m + 4 + m - 1 + 1 - - Q - - + + Q + + - + + + - - 4 - m + + 4 + m - 1 + 1 \mathbb{Q}(\sqrt{4m+1}) [/itex] 

## Formula 24 ¶

### Original ¶

$1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  1 1 1 1 [/itex] 

### Canonicalized ¶

$1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  1 1 1 1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + 1 - 1 + 1 - 1 + 1 1 [/itex] 

## Formula 25 ¶

### Original ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Canonicalized ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + m - m + m - m + m m [/itex] 

## Formula 26 ¶

### Original ¶

$4m+1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  4 m + 1 4 m 1 + 4 m 1 4m+1 [/itex] 

### Canonicalized ¶

$4m+1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  4 m + 1 4 m 1 + 4 m 1 4m+1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 @@ -1,39 +1,38 @@ - - + + 4 - m + 1 - 4 - m + 4 + m - 1 + 1 - + + + - - 4 - m + + 4 + m - 1 + 1 4m+1 [/itex] 

## Formula 27 ¶

### Original ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Canonicalized ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,26 +1,26 @@ - - + + f m - subscript - f - m + subscript + f + m - - f - m + + f + m f_{m} [/itex] 

## Formula 28 ¶

### Original ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Canonicalized ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + m - m + m - m + m m [/itex] 

## Formula 29 ¶

### Original ¶

$4m+1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  4 m + 1 4 m 1 + 4 m 1 4m+1 [/itex] 

### Canonicalized ¶

$4m+1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  4 m + 1 4 m 1 + 4 m 1 4m+1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 @@ -1,39 +1,38 @@ - - + + 4 - m + 1 - 4 - m + 4 + m - 1 + 1 - + + + - - 4 - m + + 4 + m - 1 + 1 4m+1 [/itex] 

## Formula 30 ¶

### Original ¶

$m\geq 0$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  m 0 m 0 m 0 m\geq 0 [/itex] 

### Canonicalized ¶

$m\geq 0$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  m 0 m 0 m 0 m\geq 0 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,27 +1,27 @@ - - + + m 0 - m - 0 + m + 0 - - m - 0 + + m + 0 m\geq 0 [/itex] 

## Formula 31 ¶

### Original ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Canonicalized ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,26 +1,26 @@ - - + + f m - subscript - f - m + subscript + f + m - - f - m + + f + m f_{m} [/itex] 

## Formula 32 ¶

### Original ¶

$4m+1=u^{2}D$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74  4 m + 1 = u 2 D 4 m 1 superscript u 2 D = + 4 m 1 u 2 D 4m+1=u^{2}D [/itex] 

### Canonicalized ¶

$4m+1=u^{2}D$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72  4 m + 1 = u 2 D 4 m 1 superscript u 2 D = + 4 m 1 u 2 D 4m+1=u^{2}D [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 @@ -1,74 +1,72 @@ - - + + 4 - m + 1 = u 2 - D - 4 - m + 4 + m - 1 + 1 - superscript - u - 2 + superscript + u + 2 - D + D - = + = - + + + - - 4 - m + + 4 + m - 1 + 1 - + - - u - 2 + + u + 2 - D + D 4m+1=u^{2}D [/itex] 

## Formula 33 ¶

### Original ¶

$D$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  D D D D [/itex] 

### Canonicalized ¶

$D$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  D D D D [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + D - D + D - D + D D [/itex] 

## Formula 34 ¶

### Original ¶

$u$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  u u u u [/itex] 

### Canonicalized ¶

$u$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  u u u u [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + u - u + u - u + u u [/itex] 

## Formula 35 ¶

### Original ¶

$D=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  D = 1 D 1 = D 1 D=1 [/itex] 

### Canonicalized ¶

$D=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  D = 1 D 1 = D 1 D=1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,27 +1,27 @@ - - + + D = 1 - D - 1 + D + 1 - = - D - 1 + = + D + 1 D=1 [/itex] 

## Formula 36 ¶

### Original ¶

$1

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63  1 < D < m 1 / 12 1 D superscript m 1 12 1 < D < m / 1 12 1<D<m^{1/12} [/itex] 

### Canonicalized ¶

$1

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63  1 < D < m 1 / 12 1 D superscript m 1 12 1 < D < m / 1 12 1<D<m^{1/12} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 @@ -1,63 +1,63 @@ - - + + 1 < D < m 1 / 12 - 1 - D + 1 + D - - - superscript - m - - - 1 - 12 + + + superscript + m + + + 1 + 12 - - 1 - < - D - < - - - m + + 1 + < + D + < + + + m - / - 1 - 12 + / + 1 + 12 1<D<m^{1/12} [/itex] 

## Formula 37 ¶

### Original ¶

$D\geq m^{1/12}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50  D m 1 / 12 D superscript m 1 12 D m / 1 12 D\geq m^{1/12} [/itex] 

### Canonicalized ¶

$D\geq m^{1/12}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50  D m 1 / 12 D superscript m 1 12 D m / 1 12 D\geq m^{1/12} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 @@ -1,50 +1,50 @@ - - + + D m 1 / 12 - D + D - superscript - m + superscript + m - 1 - 12 + 1 + 12 - - D + + D - - m + + m - / - 1 - 12 + / + 1 + 12 D\geq m^{1/12} [/itex] 

## Formula 38 ¶

### Original ¶

$4m+1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  4 m + 1 4 m 1 + 4 m 1 4m+1 [/itex] 

### Canonicalized ¶

$4m+1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  4 m + 1 4 m 1 + 4 m 1 4m+1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 @@ -1,39 +1,38 @@ - - + + 4 - m + 1 - 4 - m + 4 + m - 1 + 1 - + + + - - 4 - m + + 4 + m - 1 + 1 4m+1 [/itex] 

## Formula 39 ¶

### Original ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Canonicalized ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,26 +1,26 @@ - - + + f m - subscript - f - m + subscript + f + m - - f - m + + f + m f_{m} [/itex] 

## Formula 40 ¶

### Original ¶

$D=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  D = 1 D 1 = D 1 D=1 [/itex] 

### Canonicalized ¶

$D=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  D = 1 D 1 = D 1 D=1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,27 +1,27 @@ - - + + D = 1 - D - 1 + D + 1 - = - D - 1 + = + D + 1 D=1 [/itex] 

## Formula 41 ¶

### Original ¶

$m=2$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  m = 2 m 2 = m 2 m=2 [/itex] 

### Canonicalized ¶

$m=2$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  m = 2 m 2 = m 2 m=2 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,27 +1,27 @@ - - + + m = 2 - m - 2 + m + 2 - = - m - 2 + = + m + 2 m=2 [/itex] 

## Formula 42 ¶

### Original ¶

$m=t^{2}+t+\frac{2t+1}{3}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89  m = t 2 + t + 2 t + 1 3 m superscript t 2 t 2 t 1 3 = m + t 2 t + 2 t 1 3 m=t^{2}+t+\frac{2t+1}{3} [/itex] 

### Canonicalized ¶

$m=t^{2}+t+\frac{2t+1}{3}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88  m = t 2 + t + 2 t + 1 3 m superscript t 2 t 2 t 1 3 = m + t 2 t + 2 t 1 3 m=t^{2}+t+\frac{2t+1}{3} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 @@ -1,89 +1,88 @@ - - + + m = t 2 + t + 2 - t + 1 3 - m + m - superscript - t - 2 + superscript + t + 2 - t + t - 2 - t + 2 + t - 1 + 1 - 3 + 3 - = - m + = + m - + + + - - t - 2 + + t + 2 - t + t - + - + + + - - 2 - t + + 2 + t - 1 + 1 - 3 + 3 m=t^{2}+t+\frac{2t+1}{3} [/itex] 

## Formula 43 ¶

### Original ¶

$D=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  D = 1 D 1 = D 1 D=1 [/itex] 

### Canonicalized ¶

$D=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  D = 1 D 1 = D 1 D=1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,27 +1,27 @@ - - + + D = 1 - D - 1 + D + 1 - = - D - 1 + = + D + 1 D=1 [/itex] 

## Formula 44 ¶

### Original ¶

$4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111  4 t 2 + 20 t 3 + 7 3 = u 2 4 superscript t 2 20 t 3 7 3 superscript u 2 = + 4 t 2 20 t 3 7 3 u 2 4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2} [/itex] 

### Canonicalized ¶

$4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109  4 t 2 + 20 t 3 + 7 3 = u 2 4 superscript t 2 20 t 3 7 3 superscript u 2 = + 4 t 2 20 t 3 7 3 u 2 4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 @@ -1,111 +1,109 @@ - - + + 4 - t 2 + 20 - t 3 + 7 3 = u 2 - 4 + 4 - superscript - t - 2 + superscript + t + 2 - 20 - t + 20 + t - 3 + 3 - 7 - 3 + 7 + 3 - superscript - u - 2 + superscript + u + 2 - = + = - + + + - - 4 + + 4 - - t - 2 + + t + 2 - + - - 20 - t + + 20 + t - 3 + 3 - - 7 - 3 + + 7 + 3 - - u - 2 + + u + 2 4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2} [/itex] 

## Formula 45 ¶

### Original ¶

$4t^{2}+4t+1<4t^{2}+\frac{20t}{3}+\frac{7}{3}<4t^{2}+8t+4$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215  4 t 2 + 4 t + 1 < 4 t 2 + 20 t 3 + 7 3 < 4 t 2 + 8 t + 4 4 superscript t 2 4 t 1 4 superscript t 2 20 t 3 7 3 4 superscript t 2 8 t 4 + 4 t 2 4 t 1 < + 4 t 2 20 t 3 7 3 < + 4 t 2 8 t 4 4t^{2}+4t+1<4t^{2}+\frac{20t}{3}+\frac{7}{3}<4t^{2}+8t+4 [/itex] 

### Canonicalized ¶

$4t^{2}+4t+1<4t^{2}+\frac{20t}{3}+\frac{7}{3}<4t^{2}+8t+4$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209  4 t 2 + 4 t + 1 < 4 t 2 + 20 t 3 + 7 3 < 4 t 2 + 8 t + 4 4 superscript t 2 4 t 1 4 superscript t 2 20 t 3 7 3 4 superscript t 2 8 t 4 + 4 t 2 4 t 1 < + 4 t 2 20 t 3 7 3 < + 4 t 2 8 t 4 4t^{2}+4t+1<4t^{2}+\frac{20t}{3}+\frac{7}{3}<4t^{2}+8t+4 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 @@ -1,215 +1,209 @@ - - + + 4 - t 2 + 4 - t + 1 < 4 - t 2 + 20 - t 3 + 7 3 < 4 - t 2 + 8 - t + 4 - 4 + 4 - superscript - t - 2 + superscript + t + 2 - 4 - t + 4 + t - 1 + 1 - - - - - 4 - - superscript - t - 2 + + + + + 4 + + superscript + t + 2 - - - - - 20 - t + + + + + 20 + t - 3 + 3 - - - 7 - 3 + + + 7 + 3 - - - - - - 4 - - superscript - t - 2 + + + + + + 4 + + superscript + t + 2 - - - 8 - t + + + 8 + t - 4 + 4 - + - + + + - - 4 + + 4 - - t - 2 + + t + 2 - - 4 - t + + 4 + t - 1 + 1 - < - - + + < + + + - - 4 + + 4 - - t - 2 + + t + 2 - + - - 20 - t + + 20 + t - 3 + 3 - - 7 - 3 + + 7 + 3 - < - - + + < + + + - - 4 + + 4 - - t - 2 + + t + 2 - - 8 - t + + 8 + t - 4 + 4 4t^{2}+4t+1<4t^{2}+\frac{20t}{3}+\frac{7}{3}<4t^{2}+8t+4 [/itex] 

## Formula 46 ¶

### Original ¶

$2t+1

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88  2 t + 1 < u < 2 t + 2 2 t 1 u 2 t 2 + 2 t 1 < u < + 2 t 2 2t+1<u<2t+2 [/itex] 

### Canonicalized ¶

$2t+1

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86  2 t + 1 < u < 2 t + 2 2 t 1 u 2 t 2 + 2 t 1 < u < + 2 t 2 2t+1<u<2t+2 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 @@ -1,88 +1,86 @@ - - + + 2 - t + 1 < u < 2 - t + 2 - 2 - t + 2 + t - 1 + 1 - u + u - - - - - - 2 - t + + + + + + 2 + t - 2 + 2 - + - + + + - - 2 - t + + 2 + t - 1 + 1 - < - u - < - - + + < + u + < + + + - - 2 - t + + 2 + t - 2 + 2 2t+1<u<2t+2 [/itex] 

## Formula 47 ¶

### Original ¶

$t$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  t t t t [/itex] 

### Canonicalized ¶

$t$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  t t t t [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + t - t + t - t + t t [/itex] 

## Formula 48 ¶

### Original ¶

$u$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  u u u u [/itex] 

### Canonicalized ¶

$u$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  u u u u [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + u - u + u - u + u u [/itex] 

## Formula 49 ¶

### Original ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Canonicalized ¶

$m$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  m m m m [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + m - m + m - m + m m [/itex] 

## Formula 50 ¶

### Original ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Canonicalized ¶

$f_{m}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  f m subscript f m f m f_{m} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,26 +1,26 @@ - - + + f m - subscript - f - m + subscript + f + m - - f - m + + f + m f_{m} [/itex] 

## Formula 51 ¶

### Original ¶

$1

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63  1 < D < m 1 / 12 1 D superscript m 1 12 1 < D < m / 1 12 1<D<m^{1/12} [/itex] 

### Canonicalized ¶

$1

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63  1 < D < m 1 / 12 1 D superscript m 1 12 1 < D < m / 1 12 1<D<m^{1/12} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 @@ -1,63 +1,63 @@ - - + + 1 < D < m 1 / 12 - 1 - D + 1 + D - - - superscript - m - - - 1 - 12 + + + superscript + m + + + 1 + 12 - - 1 - < - D - < - - - m + + 1 + < + D + < + + + m - / - 1 - 12 + / + 1 + 12 1<D<m^{1/12} [/itex] 

## Formula 52 ¶

### Original ¶

$p$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  p p p p [/itex] 

### Canonicalized ¶

$p$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  p p p p [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + p - p + p - p + p p [/itex] 

## Formula 53 ¶

### Original ¶

$p\equiv 1\;\;(\mathop{{\rm mod}}4D)$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70  p 1 ( mod 4 D ) p annotated 1 pmod 4 D p 1 4 D mod p\equiv 1\;\;(\mathop{{\rm mod}}4D) [/itex] 

### Canonicalized ¶

$p\equiv 1\;\;(\mathop{{\rm mod}}4D)$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68  p 1 ( mod 4 D ) p annotated 1 pmod 4 D p 1 4 D mod p\equiv 1\;\;(\mathop{{\rm mod}}4D) [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 @@ -1,70 +1,68 @@ - - + + p 1 - ( mod - - 4 - - D + + 4 + D ) - p + p - annotated - 1 + annotated + 1 - pmod - - - 4 - D + pmod + + + 4 + D - - p + + p - - 1 - - - - - - 4 - D + + 1 + + + + + + 4 + D - - mod - + + mod + p\equiv 1\;\;(\mathop{{\rm mod}}4D) [/itex] 

## Formula 54 ¶

### Original ¶

$(p,m)=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43  ( p , m ) = 1 p m 1 = p m 1 (p,m)=1 [/itex] 

### Canonicalized ¶

$(p,m)=1$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43  ( p , m ) = 1 p m 1 = p m 1 (p,m)=1 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 @@ -1,43 +1,43 @@ - - + + ( p , m ) = 1 - p - m + p + m - 1 + 1 - = + = - - p - m + + p + m - 1 + 1 (p,m)=1 [/itex] 

## Formula 55 ¶

### Original ¶

$p

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  p < D C p superscript D C < p D C p<D^{C} [/itex] 

### Canonicalized ¶

$p

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  p < D C p superscript D C < p D C p<D^{C} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 @@ -1,38 +1,38 @@ - - + + p < D C - p + p - superscript - D - C + superscript + D + C - < - p + < + p - - D - C + + D + C p<D^{C} [/itex] 

## Formula 56 ¶

### Original ¶

$C$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  C C C C [/itex] 

### Canonicalized ¶

$C$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  C C C C [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + C - C + C - C + C C [/itex] 

## Formula 57 ¶

### Original ¶

$D$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  D D D D [/itex] 

### Canonicalized ¶

$D$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  D D D D [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + D - D + D - D + D D [/itex] 

## Formula 58 ¶

### Original ¶

$C=5.5$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  C = 5.5 C 5.5 = C 5.5 C=5.5 [/itex] 

### Canonicalized ¶

$C=5.5$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  C = 5.5 C 5.5 = C 5.5 C=5.5 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,27 +1,27 @@ - - + + C = 5.5 - C - 5.5 + C + 5.5 - = - C - 5.5 + = + C + 5.5 C=5.5 [/itex] 

## Formula 59 ¶

### Original ¶

$D_{0}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  D 0 subscript D 0 D 0 D_{0} [/itex] 

### Canonicalized ¶

$D_{0}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  D 0 subscript D 0 D 0 D_{0} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 @@ -1,26 +1,26 @@ - - + + D 0 - subscript - D - 0 + subscript + D + 0 - - D - 0 + + D + 0 D_{0} [/itex] 

## Formula 60 ¶

### Original ¶

$D>D_{0}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  D > D 0 D subscript D 0 > D D 0 D>D_{0} [/itex] 

### Canonicalized ¶

$D>D_{0}$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  D > D 0 D subscript D 0 > D D 0 D>D_{0} [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 @@ -1,38 +1,38 @@ - - + + D > D 0 - D + D - subscript - D - 0 + subscript + D + 0 - > - D + > + D - - D - 0 + + D + 0 D>D_{0} [/itex] 

## Formula 61 ¶

### Original ¶

$p

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62  p < m 1 / 2 / 6 p superscript m 1 2 6 < p / m / 1 2 6 p<m^{1/2}/6 [/itex] 

### Canonicalized ¶

$p

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62  p < m 1 / 2 / 6 p superscript m 1 2 6 < p / m / 1 2 6 p<m^{1/2}/6 [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 @@ -1,62 +1,62 @@ - - + + p < m 1 / 2 / 6 - p + p - superscript - m + superscript + m - 1 - 2 + 1 + 2 - 6 + 6 - < - p + < + p - / + / - - m + + m - / - 1 - 2 + / + 1 + 2 - 6 + 6 p<m^{1/2}/6 [/itex] 

## Formula 62 ¶

### Original ¶

$p$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  p p p p [/itex] 

### Canonicalized ¶

$p$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  p p p p [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + p - p + p - p + p p [/itex] 

## Formula 63 ¶

### Original ¶

$p$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  p p p p [/itex] 

### Canonicalized ¶

$p$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  p p p p [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + p - p + p - p + p p [/itex] 

## Formula 64 ¶

### Original ¶

$n$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  n n n n [/itex] 

### Canonicalized ¶

$n$

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  n n n n [/itex] 

### Diff ¶

###### (show/hide code)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 @@ -1,15 +1,15 @@ - - + + n - n + n - n + n n [/itex] 

## Formula 65 ¶

### Original ¶

$x-\frac{1+u\sqrt{D}}{2}$