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.
The equation x + x^2 + 3*y + y^2 = 2006 has real solutions at x = (-1 - Sqrt[8025 - 12*y - 4*y^2])/2 and x = (-1 + Sqrt[8025 - 12*y - 4*y^2])/2
But for real x, we must have 8025 - 12*y - 4*y^2 >0 which is only true for y in (-46, 43).
No integers y in this range give an integer value for x.
As no integer solutions exist, 2006 is definite.
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Posted by goFish
on 2006-02-27 12:29:29 |
Solution
