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On some Diophantine equations. III. (English) Zbl 1107.11021

It is known, that equation \(m^4-n^4=7y^2\) has infinitely many solutions. The author proves that the equation \[ c_k(f^4+42f^2g^2+49g^4)+28d_k(f^3g+7fg^3)=m^2 \] where \((c_k,d_k)\) are solutions of the Pellian equation \(u^2-7v^2=1\) also has infinitely many solutions.
The proposition of the paper states that the only case, when an integer solution \((m,n,y)\) of \(m^4-n^4=7y^2\) gives an integer solution of the above equation is \(k \equiv3 \pmod 4\).
The arguments are elementary but interesting. Unfortunately there are no comments where the equations in question come from.
Part II, cf. An. Stiint. Univ. “Ovidius” Constanta, Ser. Mat. 10, No. 2, 79–86 (2002; Zbl 1084.11504).

MSC:

11D25 Cubic and quartic Diophantine equations
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