TY - JOUR

T1 - The equation (w + x + y + z) (1/w + 1/x + 1/y + 1/z) = n

AU - Bremner, Andrew

AU - Xuan, Tho Nguyen

N1 - Publisher Copyright:
© 2018 World Scientific Publishing Company.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Compos. 61(203) (1993) 117-130] investigated the Diophantine problem of representing integers n in the form (x + y + z)(1/x + 1/y + 1/z) for rationals x,y,z. For fixed n, the equation represents an elliptic curve, and the existence of solutions depends upon the rank of the curve being positive. They observed that the corresponding equation in four variables, the title equation here (representing a surface), has infinitely many solutions for each n, and remarked that it seemed plausible that there were always solutions with positive w,x,y,z when n ≥ 16. This is false, and the situation is quite subtle. We show that there cannot exist such positive solutions when n is of the form 4m2, 4m2 + 4, where m ≢ 2 (mod 4). Computations within our range seem to indicate that solutions exist for all other values of n.

AB - Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Compos. 61(203) (1993) 117-130] investigated the Diophantine problem of representing integers n in the form (x + y + z)(1/x + 1/y + 1/z) for rationals x,y,z. For fixed n, the equation represents an elliptic curve, and the existence of solutions depends upon the rank of the curve being positive. They observed that the corresponding equation in four variables, the title equation here (representing a surface), has infinitely many solutions for each n, and remarked that it seemed plausible that there were always solutions with positive w,x,y,z when n ≥ 16. This is false, and the situation is quite subtle. We show that there cannot exist such positive solutions when n is of the form 4m2, 4m2 + 4, where m ≢ 2 (mod 4). Computations within our range seem to indicate that solutions exist for all other values of n.

KW - Diophantine representation

KW - Elliptic curve

KW - Hilbert symbol

KW - quartic surface

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U2 - 10.1142/S1793042118500768

DO - 10.1142/S1793042118500768

M3 - Article

AN - SCOPUS:85040931955

SN - 1793-0421

VL - 14

SP - 1229

EP - 1246

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 5

ER -