Parametric solutions for the quadratic Diophantine equation 2a^2 + b^2

Angeline Hayden

Angeline Hayden

Answered question

2022-04-22

Parametric solutions for the quadratic Diophantine equation 2a2+b2=2c2+d2.

Answer & Explanation

silviavalls1005r

silviavalls1005r

Beginner2022-04-23Added 11 answers

Write the equation as
2(a2c2)=d2b2
i.e. 2(a+c)(ac)=(d+b)(db)
Now d+b and db must be both even (they must have the same parity since d=(d+b)+(db)2 is an integer, and they can't be both odd because 2(a+c)(ac) is even. Then (a+c)(ac) is even, so a+c and ac are both even. We can write a+c=2xy,ac=2zw,d+b=4xz,db=2yw, so a=xy+zw,c=xyzw,d=2xz+yw,b=2xzyw, or else d+b=2yw and db=4xz, which produces a solution that is the same except for b=2xz+yw.

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