A given year p (expressed in YYYY format) is defined as "Ambiguous" if there exists at least one positive integral solution of the equation x^2 + x + y^2 + 3y = p. Otherwise, the said year (p) is a "Definite" year.

For example, 1890 A.D. was an "Ambiguous" year, since (x,y)=(34,25) corresponds to a positive integral solution of x^2 + x + y^2 + 3y = 1890.

Determine, whether 2006 A.D. is an Ambiguous Year or a Definite Year.

I found the following years (1900 - 2100) to be Ambiguous:

 

1900, 1902, 1908, 1910, 1914,

1920, 1924, 1930, 1932, 1936, 1938,

1942, 1944, 1946, 1956,

1960, 1962, 1978,

1980, 1982, 1984, 1986, 1990, 1992, 1998

2000, 2004, 2008, 2014, 2018,

2020, 2022, 2026, 2032, 2034,

2044, 2046, 2050, 2054, 2058,

2062, 2064, 2068, 2070, 2072, 2074, 2076,

2080, 2086, 2088, 2098,

2100

 

Now why are they labeled "Ambiguous"?

 

Edited for editing.

Edited on February 27, 2006, 11:22 am

Posted by Leming on 2006-02-27 11:18:20