an-early-question-for-the-new-year-how-many-rectangular-triangles-with-sides-a-b-c-N-exist-with-one-side-2019-

Question Number 46809 by MJS last updated on 31/Oct/18

an early question for the new year

how many rectangular triangles with sides

a, b, c \in N^{★} exist with one side = 2019

Answered by MrW3 last updated on 02/Nov/18

a^{2} + b^{2} = c^{2}

c a s e 1 : w i t h c = 2019

a^{2} + b^{2} = 2019^{2} = 3^{2} \times 673^{2}

l e t a = 3 p, b = 3 q

\Rightarrow p^{2} + q^{2} = 673^{2}

i t h a s o n e a n d o n l y o n e s o l u t i o n :

p = 385, q = 552

\Rightarrow a = 3 \times 385 = 1155

\Rightarrow b = 3 \times 552 = 1656

c a s e 2 : w i t h a = 2019

2019^{2} + b^{2} = c^{2}

\Rightarrow c^{2} - b^{2} = (c - b) (c + b) = 2019^{2} = 3^{2} \times 673^{2} = m \times n = 1 \times 4076361 = 3 \times 1358787 = 9 \times 452929 = 673 \times 6057

c - b = m

c + b = n

\Rightarrow b = \frac{n - m}{2}

\Rightarrow c = \frac{n + m}{2}

\Rightarrow b = \frac{4076361 - 1}{2} = 2038180

\Rightarrow c = \frac{4076361 + 1}{2} = 2038181

o r

\Rightarrow b = \frac{1358787 - 3}{2} = 679392

\Rightarrow c = \frac{1358787 + 3}{2} = 679395

o r

\Rightarrow b = \frac{452929 - 9}{2} = 226460

\Rightarrow c = \frac{452929 + 9}{2} = 226469

o r

\Rightarrow b = \frac{6057 - 673}{2} = 2692

\Rightarrow c = \frac{6057 + 673}{2} = 3365

s o t o t a l l y t h e r e e x i s t 5 s u c h t r i a n g l e s :

a / b / c = 1155 / 1656 / 2019

a / b / c = 2019 / 2692 / 3365

a / b / c = 2019 / 226460 / 226469

a / b / c = 2019 / 679392 / 679395

a / b / c = 2019 / 2038180 / 2038181

Commented by MJS last updated on 02/Nov/18

{you}^{'} re right, thank you!

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