Squares in blocks from an arithmetic progression and Galois group of Laguerre polynomials

Abstract

We investigate when a product of t ≥ 2 terms taken from a set of k successive terms in arithmetic progression is a perfect square. Further, we study the Galois group of Laguerre polynomials. For this, we consider products, which are perfect squares and having terms from two blocks of arithmetic progression.

On the Galois groups of generalized Laguerre Polynomials

“…Hence we may suppose that n > 13. Further we can take n ≤ 1325 by [SaSh15,Theorem 1.4]. It suffices to prove that G(x, u, v) has Galois group S n .…”

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

“…Saradha and Shorey [63], Hanrot, Saradha and Shorey [46] and Bennett [5] together proved that for $$d=K-k=b_0=1$$, $$b_\ell =0$$ the only solutions of () are given by $$4!/3=2^3$$, $$6!/5 = 12^2$$, $$10!/7 = 720^2$$. For other papers with $$K-k=1$$, $$b_\ell =0$$ see [23, 64, 65, 67, 69]. Hajdu and Papp [40] proved that Equation () with $$K-k=1$$, $$K \geqslant 8$$ has only finitely many solutions $$x,y,\ell$$.…”

“…Saradha and Shorey [63], Hanrot, Saradha and Shorey [46] and Bennett [5] together proved that for 𝑑 = 𝐾 − 𝑘 = 𝑏 0 = 1, 𝑏 𝓁 = 0 the only solutions of (5) are given by 4!∕3 = 2 3 , 6!∕5 = 12 2 , 10!∕7 = 720 2 . For other papers with 𝐾 − 𝑘 = 1, 𝑏 𝓁 = 0 see [23,64,65,67,69]. Hajdu and Papp [40] proved that Equation (5) with 𝐾 − 𝑘 = 1, 𝐾 ⩾ 8 has only finitely many solutions 𝑥, 𝑦, 𝓁.…”

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

“…Saradha and Shorey [61], Hanrot, Saradha and Shorey [44] and Bennett [5] together proved that for d = K − k = b 0 = 1, b ℓ = 0 the only solutions of ( 5) are given by 4!/3 = 2 3 , 6!/5 = 12 2 , 10!/7 = 720 2 . For other papers with [21], [62], [63], [65], [67]. Hajdu and Papp [38] proved that equation ( 5) with K − k = 1, K ≥ 8 has only finitely many solutions x, y, ℓ.…”