A362593 Number of coprime positive integer S-unit solutions to a + b = c where a <= b < c, and where S = {prime(1), ..., prime(n)}.

0, 1, 4, 17, 63, 190, 545, 1433, 3649, 8828, 20015, 44641, 95358, 199081, 412791, 839638, 1663449

Offset: 0

Author

Robin Visser, Apr 26 2023

Keywords

• nonn
• more

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).

Examples

			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.
		

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 (Case a = 1), A361661, A362567.

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 (sols + 1)/3

Formula

a(n) = (A362567(n) + 3)/6 if n > 0.