Sunday, 20 September 2015

natural numbers - solve equation in positive integers



Can anybody help me with this equation?



Solve in N:
3x27y2+1=0


One solution is the pair (3,2), and i think this is the only pair of positive integers that can be a solution. Any idea?


Answer



There are infinitely many solutions in positive integers. 7y23x2=1 is an example of a "Pell equation", and there are standard methods for finding solutions to Pell equations.



For example, the fact that (x,y)=(2,3) is a solution to 7y23x2=1 is equivalent to noting that (27+33)(2733)=1. The fundamental unit in Q(21) is 55+1221; in particular, (55+1221)(551221)=1. Consequently, if we calculate (27+33)(55+1221)=2187+3333, it follows that (2187+3333)(21873333)=1, or 7218233332=1.




You can get infinitely many solutions (xn,yn) to 7y23x2=1 by expanding (27+33)(55+1221)n=yn7+xn3. Your solution corresponds to n=0, while the previous paragraph is n=1; for example, n=2 yields x2=36627 and y2=23978.


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