A361661 -id:A361661 - cp's OEIS frontend

A359480 Number of Q-isomorphism classes of elliptic curves E/Q with good reduction away from 2 and prime(n).

24, 752, 280, 288, 232, 336, 256, 336, 256, 296, 280, 240, 176, 168, 136, 296, 304, 176, 112, 288, 136, 304, 176, 192, 152, 216, 104, 240, 160, 144, 280, 160, 152, 168, 112, 128, 136, 232, 144, 184, 128, 152, 80, 88, 112, 112, 112, 280, 112, 288, 160, 120, 168, 112, 224, 112, 120, 112, 136

Offset: 1

Robin Visser, Mar 31 2023

nonn

R. von Känel and B. Matschke conjecture that a(n) <= a(2) = 752 for all n.

			For n = 1, there are a(1) = 24 elliptic curves over Q with good reduction outside 2, classified by Ogg (1966), with j-invariants given in A332545.
For n = 2, there are a(2) = 752 elliptic curves over Q with good reduction outside {2,3}, classified independently by Coghlan (1967) and Stephens (1965).

N. M. Stephens, The Birch Swinnerton-Dyer Conjecture for Selmer curves of positive rank, Ph.D. Thesis (1965), The University of Manchester.

F. B. Coghlan, Elliptic Curves with Conductor N = 2^m 3^n, Ph.D. Thesis (1967), The University of Manchester.
J. E. Cremona and M. P. Lingham, Finding all elliptic curves with good reduction outside a given set of primes, Experiment. Math. 16 (2007), no. 3, 303-312.
A. P. Ogg, Abelian curves of 2-power conductor, Proc. Cambridge Philos. Soc. 62 (1966), 143-148.
R. von Känel and B. Matschke, List of all rational elliptic curves with good reduction outside {2, p} up to Q-isomorphisms, 2015.

Cf. A332545, A361661.

# This is very slow!
def a(n):
    S = list(set([2, Primes()[n-1]]))
    EC = EllipticCurves_with_good_reduction_outside_S(S)
    return len(EC)

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

Original entry on oeis.org

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

Offset: 0

Robin Visser, Apr 26 2023

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).

			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.

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 (Case a = 1), A361661, A362567.

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

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

A363793 Number of Q-isomorphism classes of elliptic curves E/Q with good reduction away from prime(n).

Original entry on oeis.org

24, 8, 0, 4, 12, 0, 8, 8, 0, 0, 0, 16, 0, 6, 2, 2, 0, 2, 4, 0, 4, 4, 2, 6, 0, 2, 0, 0, 2, 4, 0, 2, 0, 2, 0, 0, 2, 4, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 10, 0, 0, 0, 2, 0, 2, 0, 4, 6, 0, 2, 0, 0, 2, 4, 0, 0, 0, 0, 6, 4, 0, 8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 8

Offset: 1

Robin Visser, Jun 22 2023

nonn

R. von Känel and B. Matschke conjecture that a(n) <= 24 for all n.

			For n = 1, there are a(1) = 24 elliptic curves over Q with good reduction outside 2, classified by Ogg (1966), with j-invariants given in A332545.
For n = 2, there are a(2) = 8 elliptic curves over Q with good reduction outside 3. A set of 8 Weierstrass equations for these curves can be given as: y^2 + y = x^3 - 270x - 1708, y^2 + y = x^3 - 30x + 63, y^2 + y = x^3 - 7, y^2 + y = x^3, y^2 + y = x^3 - 1, y^2 + y = x^3 + 20, y^2 + y = x^3 - 61, and y^2 + y = x^3 + 2.
For n = 3, Edixhoven-Groot-Top proved there are no elliptic curves over Q with good reduction away from 5, so a(3) = 0.

M. A. Bennett and A. Rechnitzer, Computing elliptic curves over Q: bad reduction at one prime, In: Melnik, R., Makarov, R., Belair, J. (eds) Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol 79. Springer, New York, NY.
B. Edixhoven, A. de Groot, and J. Top, Elliptic curves over the rationals with bad reduction at only one prime, Math. Comp. 54 (1990), no.189, 413-419.
A. P. Ogg, Abelian curves of 2-power conductor, Proc. Cambridge Philos. Soc. 62 (1966), 143-148.
B. Setzer, Elliptic curves of prime conductor, J. London Math. Soc. (2)10(1975), 367-378.
R. von Känel and B. Matschke, List of certain finite sets S of rational primes together with the number of rational elliptic curves up to rational isomorphisms with good reduction outside S, 2016.

Cf. A000040, A110620, A332545, A359480, A361661.

def a(n):
    EC = EllipticCurves_with_good_reduction_outside_S([Primes()[n-1]])
    return len(EC)

a(n) = A110620(prime(n)) + A110620(prime(n)^2) for all n > 2.

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

Original entry on oeis.org

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.

A368080 Number of Qbar-isomorphism classes of elliptic curves E/Q with good reduction outside the first n prime numbers.

Original entry on oeis.org

0, 5, 83, 442, 2140, 8980, 34960, 124124, 418816

Offset: 0

Robin Visser, Dec 10 2023

Two elliptic curves are isomorphic over Qbar (the algebraic numbers) if and only if they share the same j-invariant, thus a(n) is also the number of distinct j-invariants of elliptic curves E/Q with good reduction outside the first n prime numbers.

			For n = 0, Tate proved there are no elliptic curves over Q with good reduction everywhere, so a(0) = 0.
For n = 1, there are 24 elliptic curves over Q with good reduction outside 2, classified by Ogg (1966). These are divided into a(1) = 5 Qbar-isomorphism classes, where the 5 corresponding j-invariants are given by 128, 1728, 8000, 10976, and 287496 (sequence A332545).

N. M. Stephens, The Birch Swinnerton-Dyer Conjecture for Selmer curves of positive rank, Ph.D. Thesis (1965), The University of Manchester.

M. A. Bennett, A. Gherga, and A. Rechnitzer, Computing elliptic curves over Q, Math. Comp., 88(317):1341-1390, 2019.
A. J. Best and B. Matschke, Elliptic curves with good reduction outside {2, 3, 5, 7, 11, 13}.
A. J. Best and B. Matschke, Elliptic curves with good reduction outside of the first six primes, arXiv:2007.10535 [math.NT], 2020.
F. B. Coghlan, Elliptic Curves with Conductor N = 2^m 3^n, Ph.D. Thesis (1967), The University of Manchester.
B. Matschke, Elliptic curve tables.
A. P. Ogg, Abelian curves of 2-power conductor, Proc. Cambridge Philos. Soc. 62 (1966), 143-148.
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. A332545, A361661.

# This is very slow for n > 2
def a(n):
    S = Primes()[:n]
    EC = EllipticCurves_with_good_reduction_outside_S(S)
    return len(set(E.j_invariant() for E in EC))

cp's OEIS Frontend

A359480 Number of Q-isomorphism classes of elliptic curves E/Q with good reduction away from 2 and prime(n).

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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)}.

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A363793 Number of Q-isomorphism classes of elliptic curves E/Q with good reduction away from prime(n).

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

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SageMath

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A368080 Number of Qbar-isomorphism classes of elliptic curves E/Q with good reduction outside the first n prime numbers.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Sage