A362567 Number of rational solutions to the S-unit equation x + y = 1, where S = {prime(i): 1 <= i <= n}.

0, 3, 21, 99, 375, 1137, 3267, 8595, 21891, 52965, 120087, 267843, 572145, 1194483, 2476743, 5037825, 9980691

Offset: 0

Author

Robin Visser, Apr 25 2023

Keywords

• nonn
• more

Comments

Let S = {p_1, p_2, ..., p_n} be a finite set of prime numbers. A rational S-unit is a rational number x such that abs(x) = p_1^k_1 * p_2^k_2 * ... * p_n^k_n for some integers k_1, k_2, ..., k_n.

Thus a(n) is the number of ordered pairs (x,y) of rational numbers such that x+y=1 and v_p(x) = v_p(y) = 0 for all primes p greater than prime(n), i.e., the primes dividing the numerator or denominator of x or y are some subset of the first n prime numbers.

Mahler (1933) first proved that a(n) is finite for all n, with effective bounds first given by Györy (1979).

Examples

			For n = 1, the a(1) = 3 solutions are -1 + 2 = 1, 1/2 + 1/2 = 1, and 2 + -1 = 1.
For n = 2, the a(2) = 21 solutions are -8 + 9 = 1, -3 + 4 = 1, -2 + 3 = 1, -1 + 2 = 1, -1/2 + 3/2 = 1, -1/3 + 4/3 = 1, -1/8 + 9/8 = 1, 1/9 + 8/9 = 1, 1/4 + 3/4 = 1, 1/3 + 2/3 = 1, 1/2 + 1/2 = 1, 2/3 + 1/3 = 1, 3/4 + 1/4 = 1, 8/9 + 1/9 = 1, 9/8 + -1/8 = 1, 4/3 + -1/3 = 1, 3/2 + -1/2 = 1, 2 + -1 = 1, 3 + -2 = 1, 4 + -3 = 1, and 9 + -8 = 1.
		

Links

• A. Alvarado, A. Koutsianas, B. Malmskog, C. Rasmussen, C. Vincent, and M. West, A robust implementation for solving the S-unit equation and several applications, arXiv:1903.00977 [math.NT], 2019.
• B. M. M. de Weger, Solving exponential Diophantine equations using lattice basis reduction algorithms, J. Number Theory 26 (1987), no. 3, 325-367.
• R. von Känel and B. Matschke, Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture, arXiv:1605.06079 [math.NT], 2016.

Crossrefs

Cf. A002071, A361661, A362593.

Programs

SageMath

from sage.rings.number_field.S_unit_solver import solve_S_unit_equation
def a(n):
    Q = CyclotomicField(1)
    S = Q.primes_above(prod([p for p in Primes()[:n]]))
    sols = len(solve_S_unit_equation(Q, S))
    return 2*sols - 1

Formula

a(n) = 6*A362593(n) - 3 if n > 0.