A376243 Nonnegative integers N = x*y*z = x+y+z for some rational x, y, z.

0, 6, 7, 9, 13, 14, 15, 16, 19

Offset: 1

Author

M. F. Hasler, Sep 16 2024

Keywords

• nonn
• more

Comments

Obviously all of x, y and z must be nonzero for all solutions N > 0. For any N = x*y*z = x+y+z, one gets -N from (-x, -y, -z), so considering only N >= 0 is not a restriction. Either none or exactly two among x, y and z must be negative.

For given N, the problem amounts to finding fractions x and y such that x*y^2 + x*(x - N)*y + N = 0, which in turn corresponds to finding rational points on the elliptic curve Y^2 = X^3 + N^2*(X+4)^2 (with X = -4*N/x and Y = 4*N*D/x^2, where D^2 is the discriminant of the previous quadratic in y).

It appears that (for N > 0) we have a rational solution iff this elliptic curve has nonzero rank. (Is there any counterexample?) If so, the sequence goes (0, 6, 7, 9, 13, 14, 15, 16, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 37, 38, 40, 43, 44, 45, 46, 48, 49, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 86, 87, ...)

Examples

			The first few terms correspond to the following solutions (|x| <= |y| <= |z|):
    N  |    x    |    y    |    z
  -----+---------+---------+---------
    0  |    0    |    0    |    0    (or any rational y = -z).
    6  |    1    |    2    |    3    (and also {25/21, 54/35, 49/15}).
    7  |   7/6   |   4/3   |   9/2
    9  |   1/2   |    4    |   9/2
   13  |  36/77  | 121/42  | 637/66
   14  |   1/3   |    9    |  14/3
   15  |   1/2   |   5/2   |   12
   16  |  -2/3   |  -4/3   |   18
   19  | 121/234 | 324/143 |3211/198
  ...
All terms of A054000 (2*n^2-2: 0, 6, 16, 30, 48, 70, 96, 126, 160, 198, ...) are in the sequence, as product and sum of the triple (2*n^2, 1/n - 1, -1/n - 1).
		

Links

• Allan MacLeod, Elliptic Curves in Recreational Number Theory, arXiv:1610.03430 [math.NT], Oct. 2016.
• Victor Miller and others, in reply to Keith F. Lynch, Re: Integer sums and products, math-fun mailing list (available for subscribers), Sep. 2024.

Crossrefs

Cf. A376241-A376242 for an enumeration of all possible solutions (not in the order of increasing N) using the Stern-Brocot sequence A002487.

A054000 (2*n^2-2) is a subsequence.

Programs

PARI

select( {is_A376243(n)=!n||ellrank(ellinit([0,1,0,8,16]*n^2))}, [0..30]) \\ Assuming there's a rational solution iff the elliptic curve has rank > 0. - M. F. Hasler, Sep 23 2024