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A362593
Number of coprime positive integer S-unit solutions to a + b = c where a <= b < c, and where S = {prime(1), ..., prime(n)}.
2
0, 1, 4, 17, 63, 190, 545, 1433, 3649, 8828, 20015, 44641, 95358, 199081, 412791, 839638, 1663449
OFFSET
0,3
COMMENTS
Let S = {p_1, p_2, ..., p_n} be a finite set of prime numbers. A positive integer S-unit is a positive integer x such that x = p_1^k_1 * p_2^k_2 * ... * p_n^k_n for some nonnegative integers k_1, k_2, ..., k_n.
Thus a(n) is the number of positive integer triples (a,b,c) such that a + b = c, gcd(a,b,c) = 1, a <= b < c and v_p(a) = v_p(b) = v_p(c) = 0 for all primes p greater than prime(n), i.e., the primes dividing a, b or c 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) = (A362567(n) + 3)/6 if n > 0.
EXAMPLE
For n = 2, the a(2) = 4 solutions are 1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4, and 1 + 8 = 9.
For n = 3, the a(3) = 17 solutions are 1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5, 1 + 5 = 6, 1 + 8 = 9, 1 + 9 = 10, 1 + 15 = 16, 1 + 24 = 25, 1 + 80 = 81, 2 + 3 = 5, 2 + 25 = 27, 3 + 5 = 8, 3 + 125 = 128, 4 + 5 = 9, 5 + 27 = 32, and 9 + 16 = 25.
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 (sols + 1)/3
CROSSREFS
Sequence in context: A286210 A339286 A252815 * A191272 A122231 A119916
KEYWORD
nonn,more
AUTHOR
Robin Visser, Apr 26 2023
STATUS
approved