Can anybody help me with this equation?
Solve in N:
3x2−7y2+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. 7y2−3x2=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 7y2−3x2=1 is equivalent to noting that (2√7+3√3)(2√7−3√3)=1. The fundamental unit in Q(√21) is 55+12√21; in particular, (55+12√21)(55−12√21)=1. Consequently, if we calculate (2√7+3√3)(55+12√21)=218√7+333√3, it follows that (218√7+333√3)(218√7−333√3)=1, or 7⋅2182−3⋅3332=1.
You can get infinitely many solutions (xn,yn) to 7y2−3x2=1 by expanding (2√7+3√3)(55+12√21)n=yn√7+xn√3. 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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