Parameterized norm form equations with arithmetic progressions

Buchmann and Pethő [5] observed that following algebraic integer 10 + 9α + 8α + 7α + 6α + 5α + 4α, with α = 3 is a unit. Since the coefficients form an arithmetic progressions they have found a solution to the Diophantine equation (1) NK/Q(x0 + αx1 + · · ·+ x6α) = ±1, such that (x0, . . . , x6) ∈ Z is an arithmetic progression. Recently Bérczes and Pethő [3] considered the Diophantine equation (2) NK/Q(x0 + αx1 + · · ·+ xn−1αn−1) = m, where α is an algebraic number of degree n, K = Q(α), m ∈ Z and (x0, x1, . . . , xn−1) ∈ Z is nearly an arithmetic progression. The sequence (x0, . . . , xn−1) is said to be nearly an arithmetic progression if there exists d ∈ Z and 0 < δ ∈ R such that |(xi − xi−1)− d| ≤ (max{|x0|, . . . , |xn−1|})1−δ, (i = 1, . . . , n− 1). They proved that equation (2) has only finitely many solutions provided β := nα n αn−1 − α α−1 is an algebraic number of degree at least 3. Moreover they showed that the solution found by Buchmann and Pethő is the only solution to (1). Bérczes and Pethő also considered arithmetic progressions arising from the norm equation (2), where α is a root of X − a, with n ≥ 3 and 2 ≤ a ≤ 100 (see [2]). Let fa ∈ Z[X] be the Thomas polynomial fa := X − (a− 1)X − (a + 2)X − 1. The aim of this paper is to prove the following theorem: Theorem 1. Let α be a root of the polynomial fa, with a ∈ Z. Then the only solutions to the norm form inequality (3) ∣NK/Q(x0 + x1α + x2α) ∣∣ ≤ |2a + 1| such that x0 < x1 < x2 is an arithmetic progression and (x1, x2, x3) is primitive are either (x1, x2, x3) = (−2,−1, 0), (−1, 0, 1) and (0, 1, 2), or they are sporadic solutions that are listed in table 1.