Prove that there are no integers $x,y$ such that $y^2=x^3-73$. Thank you.
Prove that the equation $y^2=x^3-73$ has no integer solutions
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number-theory
algebraic-number-theory
elliptic-curves
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1$(2a+1)^2+73=2(2a^2+2a+37)$ even but not divisible by $8,$ so $y$ must be even. – 2012-12-10
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6Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove") to be rude when asking for help; please consider rewriting your post. – 2012-12-10
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7Equations of the form $y^2=x^3-k$ can sometimes be solved by purely elementary considerations, sometimes by factorization in the field with $\sqrt{-k}$, sometimes by factorization in the field with $\root3\of k$, and sometimes more advanced methods are needed. That's one reason it would be helpful to know the context in which you encountered the problem. – 2012-12-10
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6Why did this question get 5 upvotes? Bad manners, no context or trace of effort on the poster's part. – 2013-04-30