Pseudo-Heronian triangles whose squares of the lengths of one or two sides are prime numbers
Articles
Edmundas Mazėtis
Vilnius University
https://orcid.org/0000-0001-8604-9179
Grigorijus Melničenko
Vytauto Magnus University
Published 2021-12-20
https://doi.org/10.15388/LMR.2021.25231
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Keywords

Heronian triangles
pseudo-Heronian triangles
prime numbers
sum of squares of two numbers

How to Cite

Mazėtis E. and Melničenko G. (2021) “Pseudo-Heronian triangles whose squares of the lengths of one or two sides are prime numbers”, Lietuvos matematikos rinkinys, 62(B), pp. 80-85. doi: 10.15388/LMR.2021.25231.

Abstract

The authors introduced the concept of a pseudo-Heron triangle, such that squares of sides are integers, and the area is an integer multiplied by $2$. The article investigates the case of pseudo-Heron triangles such that the squares of the two sides of the pseudo-Heron triangle are primes of the form $4k+1$. It is proved that for any two predetermined prime numbers of the form $4k+1$ there exist pseudo-Heron triangles with vertices on an integer lattice, such that these two primes are the sides of these triangles and such triangles have a finite number. It is also proved that for any predetermined prime number of the form $4k+1$, there are isosceles triangles with vertices on an integer lattice, such that this prime is equal to the values of two sides and there are only a finite number of such triangles.

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