Cubes in products of terms from an arithmetic progression

The Diophantine equation f(x)=g(y)$f(x)=g(y)$ for polynomials with simple rational roots

“…All solutions of Equation () with $$K-k=2$$, $$k \geqslant 4$$, $$\ell \geqslant 3$$ have been given in [55] and [68]. For papers with $$K-k \geqslant 2, b_{\ell }=0$$ see [3] and [22]. Hajdu, Papp and Tijdeman [41] provided effective upper bounds for $$\max (|x|,|y|,\ell )$$ in () under the assumption that $$K-k < cK^{2/3}$$ for some explicit $$c>0$$.…”

“…All solutions of Equation ( 5) with 𝐾 − 𝑘 = 2, 𝑘 ⩾ 4, 𝓁 ⩾ 3 have been given in [55] and [68]. For papers with 𝐾 − 𝑘 ⩾ 2, 𝑏 𝓁 = 0 see [3] and [22]. Hajdu, Papp and Tijdeman [41] provided effective upper bounds for max(|𝑥|, |𝑦|, 𝓁) in ( 5) under the assumption that 𝐾 − 𝑘 < 𝑐𝐾 2∕3 for some explicit 𝑐 > 0.…”

The Diophantine equation $f(x)=g(y)$ for polynomials with simple rational roots

“…Mukhopadhyay and Shorey [53] for ℓ = 2 and Saradha and Shorey [66] for ℓ ≥ 3 determined all solutions of equation ( 5) with [3] and [20]. Hajdu, Papp and Tijdeman [39] provided effective upper bounds for max(|x|, |y|, ℓ) in ( 5) under the assumption that K − k < cK 2/3 for some explicit c > 0.…”

“…We have 4b 3 = 4 • 5 3 = 4 2 + 22 2 = 10 2 +20 2 . Taking (w 1 , w 2 ) = (4, 22),(10,20), by(24), we get (w 3 , w 4 ) = (−4, −22), (−10, −20), and by(26), u = −23506, 8750, respectively. That is, we haveD 4 (x, 5 3 ) = (x + 4)(x − 4)(x + 22)(x − 22) + 23506 = = (x + 10)(x − 10)(x + 20)(x − 20) − 8750.Since, by(32), the equation F (x) := D 4 (x, 5 3 ) = D 3 (y, 5 4 ) =: G(y) has infinitely many integral solutions (x, y) = (D 3 (X, 5), D 4 (X, 5)) (X ∈ Z), we obtain that the equationf (x) := (x+4)(x−4)(x+22)(x−22)(x+10)(x−10)(x+20)(x−20) = = (D 3 (y, 5 4 ) − 23506)(D 3 (y, 5 4 ) + 8750) =: g(y) has the same solutions.…”