A362593 -id:A362593 - cp's OEIS frontend

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

Robin Visser, Apr 25 2023

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).

			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.

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.

Cf. A002071, A361661, A362593.

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

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

A376925 a(n) is the largest number that can be written as x + y with x and y coprime and such that each of x, y, and x + y are prime(n)-smooth.

Original entry on oeis.org

2, 9, 128, 4375, 18225, 1771561, 27295744, 1375325568, 6313843404, 289478389760

Offset: 1

Zhicheng Wei, Oct 10 2024

The terms given here can be verified by checking that the number of solutions up to a(n) equals A362593(n) and a(n) is a solution x+y.
a(2) = 9 corresponds to Catalan's conjecture (Mihăilescu's theorem).
a(4) = 4375 corresponds to the final term of A303332.

			a(3) = 128 because prime(3) = 5, and 125 + 3 = 128 with 125 and 3 coprime, and 125, 3 and 128 are all 5-smooth numbers, and no number larger than 128 has these properties.
Table x + y = a(n) is shown below (q gives abc triple quality):
  n=1: 1 + 1 = 2 (q=1),
  n=2: 8 + 1 = 9 (q=1.226)
  n=3: 125 + 3 = 128 (q=1.426)
  n=4: 4374 + 1 = 4375 (q=1.567)
  n=5: 14641 + 3584 = 18225 (q=1.267)
  n=6: 1771470 + 91 = 1771561 (q=1.395)
  n=7: 27217619 + 78125 = 27295744 (q=1.421)
  n=8: 1371299293 + 4026275 = 1375325568 (q=1.31)
  n=9: 4867359029 + 1446484375 = 6313843404 (q=1.17)
  n=10: 289478257991 + 131769  = 289478389760 (q=1.16)

Wikipedia, Abc conjecture.
Wikipedia, Catalan's conjecture.

Cf. A303332, A362593.

a(7)-a(8) from Andrew Howroyd, Oct 12 2024

a(9)-a(10) from David A. Corneth, Nov 24 2024

cp's OEIS Frontend

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

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