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.
Highlights
an Open Access article distributed under the terms of the Creative Commons Attribution Licence
Taip pat bet kuriam 4k + 1 pavidalo pirminiam skaičiui p egzistuotų tik 5 skirtingi lygiašoniai pseudo Herono trikampiai, kurių šoninių kraštinių ilgių kvadratai būtų lygūs tam pirminiam skaičiui
Summary
Kurio kraštinių ilgių kvadratai yra sveikieji skaičiai, o dviguba ploto reikšmė – irgi sveikasis skaičius, vadinamas pseudo Herono trikampiu. Kurio statinių ilgiai – sveikieji skaičiai, irgi yra atskiras pseudo Herono trikampio atvejis. Bet kuriems dviem duotiesiems 4k + 1 pavidalo pirminiams skaičiams egzistuoja lygiai aštuoni pseudo Herono trikampiai, kurių viršūnės yra sveikaskaitės gardelės taškuose, dviejų kraštinių ilgių kvadratai lygūs tiems duotiesiems skaičiams.
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