# pythagorean: Pythagorean Triples In numbers: Number-Theoretic Functions

## Description

Generates all primitive Pythagorean triples (a, b, c) of integers such that a^2 + b^2 = c^2, where a, b, c are coprime (have no common divisor) and c_1 ≤ c ≤ c_2.

## Usage

 `1` ```pythagorean_triples(c1, c2) ```

## Arguments

 `c1, c2` lower and upper limit of the hypothenuses `c`.

## Details

If (a, b, c) is a primitive Pythagorean triple, there are integers m, n with 1 ≤ n < m such that

a = m^2 - n^2, b = 2 m n, c = m^2 + n^2

with gcd(m, n) = 1 and m - n being odd.

## Value

Returns a matrix, one row for each Pythagorean triple, of the form `(m n a b c)`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```pythagorean_triples(100, 200) ## [,1] [,2] [,3] [,4] [,5] ## [1,] 10 1 99 20 101 ## [2,] 10 3 91 60 109 ## [3,] 8 7 15 112 113 ## [4,] 11 2 117 44 125 ## [5,] 11 4 105 88 137 ## [6,] 9 8 17 144 145 ## [7,] 12 1 143 24 145 ## [8,] 10 7 51 140 149 ## [9,] 11 6 85 132 157 ## [10,] 12 5 119 120 169 ## [11,] 13 2 165 52 173 ## [12,] 10 9 19 180 181 ## [13,] 11 8 57 176 185 ## [14,] 13 4 153 104 185 ## [15,] 12 7 95 168 193 ## [16,] 14 1 195 28 197 ```

### Example output

```      [,1] [,2] [,3] [,4] [,5]
[1,]   10    1   99   20  101
[2,]   10    3   91   60  109
[3,]    8    7   15  112  113
[4,]   11    2  117   44  125
[5,]   11    4  105   88  137
[6,]    9    8   17  144  145
[7,]   12    1  143   24  145
[8,]   10    7   51  140  149
[9,]   11    6   85  132  157
[10,]   12    5  119  120  169
[11,]   13    2  165   52  173
[12,]   10    9   19  180  181
[13,]   11    8   57  176  185
[14,]   13    4  153  104  185
[15,]   12    7   95  168  193
[16,]   14    1  195   28  197
```

numbers documentation built on May 15, 2021, 1:08 a.m.