Solve-for-integers-6x-5y-2-4z-x-2y-2-3z-2021-

Question Number 150558 by mathdanisur last updated on 13/Aug/21

Solve for integers :

(6 x + {5 y}^{2}) (4 z + x) ({2 y}^{2} + 3 z) = 2021

Answered by MJS_new last updated on 13/Aug/21

2021 = 43 \times 47

a b c = 2021

one factor = 1

one factor = 43

one factor = 47

(1) 6 x + 5 y^{2} = a \Rightarrow y^{2} = \frac{a - 6 x}{5}

(2) 4 z + x = b

(3) 2 y^{2} + 3 z = c \Rightarrow y^{2} = \frac{c - 3 z}{2}

================

(1) / (3) 12 x - 15 z - 2 a + 5 c = 0

(2) x + 4 z - b = 0

\Rightarrow

x = \frac{8 a + 15 b - 20 c}{63} \land z = \frac{- 2 a + 12 b + 5 c}{63}

\Rightarrow

y = \pm \sqrt{\frac{a - 6 b + 8 c}{21}}

testing all variations for a, b, c = 1, 43, 47

\Rightarrow

no integer solution

Commented by mathdanisur last updated on 14/Aug/21

Thank you Ser cool