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A362567
Number of rational solutions to the S-unit equation x + y = 1, where S = {prime(i): 1 <= i <= n}.
1
0, 3, 21, 99, 375, 1137, 3267, 8595, 21891, 52965, 120087, 267843, 572145, 1194483, 2476743, 5037825, 9980691
OFFSET
0,2
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).
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.
FORMULA
a(n) = 6*A362593(n) - 3 if n > 0.
EXAMPLE
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.
PROG
(Sage)
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
CROSSREFS
Sequence in context: A367997 A264246 A144883 * A074597 A350726 A241795
KEYWORD
nonn,more
AUTHOR
Robin Visser, Apr 25 2023
STATUS
approved