Izvje??a sa znanstvenih skupova - pregledi

摘要

Cassels [2] was challenged to determine when the sum of three consecutivecubes equals a square. He [2] reduced the problem to finding integral points on theelliptic curve y2 = 3x(x2+2). Using the arithmetic of certain quartic number fields,he obtained that the integral points on the above elliptic curve were (x, y) = (0, 0),(1, 3), (2, 6), and (24, 204).Using the classical work of Ljunggren [5] and its generalizations (see [1,4,10,11]), Luca and Walsh [6] considered the problem of finding the number of positiveinteger solutions to the Diophantine equation y2 = nx(x2 + 2), where n 1 isa positive integer. They proved that the number of positive integer solutions toy2 = nx(x2 + 2) is at most 3 · 2ω(n) ? 1, where ω(n) is the number of distinct primefactors of n. In [3], Chen considered the case where n is a prime number greaterthan 3. He proved, in particular, that the Diophantine equation y2 = nx(x2 + 2)has at most two positive integer solutions.