numbers (version 0.8-5)

pythagorean_triples: Pythagorean Triples

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 \(c_1 \le c \le c_2\).

Usage

pythagorean_triples(c1, c2)

Value

Returns a matrix, one row for each Pythagorean triple, of the form

(m n a b c).

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 \(1 \le n < m\) 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.

References

https://mathworld.wolfram.com/PythagoreanTriple.html

Examples

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