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Torsion in Groups of Integral Triangles ()

Let 0＜*γ*＜π be a fixed pythagorean angle. We study the abelian group *H _{r}* of primitive integral triangles (

*a,b,c*) for which the angle opposite side

*c*is

*γ*. Addition in

*H*is defined by adding the angles

_{r }*β*

*opposite side*

*b*and modding out by π-

*γ*. The only

*H*for which the structure is known is

_{r}*H*

_{π}

_{/}_{2}, which is free abelian. We prove that for general

*γ*,

*H*has an element of order two iff 2(1- cos

_{r}*γ*) is a rational square, and it has elements of order three iff the cubic (2cos

*γ*)

*x*

^{3}-3

*x*

^{2}+1=0 has a rational solution 0＜

*x*＜1. This shows that the set of values of

*γ*for which

*H*has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z

_{r}_{2}or Z

_{3}in

*H*. Finally, we give some examples of higher order torsion elements in

_{r}*H*.

_{r}Keywords

Share and Cite:

*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 116-120. doi: 10.4236/apm.2013.31015.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | O. Taussky, “Sums of Squares,” American Mathematical Monthly, Vol. 77, No. 8, 1970, pp. 805-830. doi:10.2307/2317016 |

[2] | E. J. Eckert, “The Group of Primitive Pythagorean Triangles,” Mathematics Magazine, Vol. 57, No. 1, 1984, pp. 22-27. doi:10.2307/2690291 |

[3] | J. Mariani, “The Group of the Pythagorean Numbers”, American Mathematical Monthly, Vol. 69, 1962, pp. 125-128. doi:10.2307/2312540 |

[4] | B. H. Margolius, “Plouffe’s Constant is Transcendental,” http://www.plouffe.fr/simon/articles/plouffe.pdf. |

[5] | E. J. Eckert and P. D. Vestergaard, “Groups of Integral Triangles,” Fibonacci Quarterly, Vol. 27, No. 5, 1989, pp. 458-464. |

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