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Solve-the-equation-x-2-x-1-2-y-2-y-1-2-2021-x-y-Z-

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Question Number 137848 by Bekzod Jumayev last updated on 07/Apr/21
Solve the equation (x^2 −x+1)^2 +(y^2 +y+1)^2 =2021 x,y∈Z
Solvetheequation(x2−x+1)2+(y2+y+1)2=2021x,y∈Z
Answered by mr W last updated on 07/Apr/21
x^2 −x=(x−1)x=even x^2 −x+1=odd (x^2 −x+1)^2 =odd^2 =odd similarly (y^2 +y+1)^2 =odd (x^2 −x+1)^2 +(y^2 +y+1)^2 =odd+odd=even but 2021 is odd, therefore (x^2 −x+1)^2 +(y^2 +y+1)^2 =2021 has no solution.
x2−x=(x−1)x=evenx2−x+1=odd(x2−x+1)2=odd2=oddsimilarly(y2+y+1)2=odd(x2−x+1)2+(y2+y+1)2=odd+odd=evenbut2021isodd,therefore(x2−x+1)2+(y2+y+1)2=2021hasnosolution.

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