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Solve for integers

1)$$ x^2 + 7 = y^3 $$

2)$$ a^2 + 5 = z^3 $$

for some integer $a,x,y$ and $z$.

For part (1), I started with the equation:

$$ x^2 + 7 = y^3 $$

I then subtracted 8 from both sides, rewriting it as:

$$ x^2 - 1 = y^3 - 2^3 $$

Recognizing a difference of cubes on the right-hand side, I factored it as:

$$ x^2 - 1 = (y - 2)(y^2 + 2y + 4) $$

From here, I found one solution: $x = 1$ and $y = 2$, but I was unsure how to proceed further.

For part (2), I didn't know how to approach the problem.

Thomas Andrews
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K41
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    You should ask one question per post – Sine of the Time Apr 02 '25 at 10:37
  • @SineoftheTime they are part of the same question and possible share a common idea. – K41 Apr 02 '25 at 10:43
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    Your equations is Mordell curves – Thinh Dinh Apr 02 '25 at 10:47
  • @ThinhDinh might be, but I think we should look it as a diophantine equation for now. – K41 Apr 02 '25 at 10:53
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    Looking at them as Mordell curves is looking at them as diophantine equations. And there is an enormous literature on Mordell curves (and many questions about them have been asked and answered on this website already – did you do a search to see whether your two have already been solved here?). The general picture is that some of them can be solved by elementary means like congruences, some of them can be solved with a little Algebraic Number Theory, some of them can be solved with a lot of Algebraic Number Theory, and some of them no one has been able to solve. – Gerry Myerson Apr 02 '25 at 11:07
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    $y^2=x^3-5$ has a complete solution in Conrad's article (see the duplicate answer), in Theorem $2.2$ and Example $3.7$. It is only using elementary methods. – Dietrich Burde Apr 02 '25 at 12:27
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    $y^2=x^3-7$ has the integer solutions $(2,±1)(2,±1), (32,±181)(32,±181)$, which follows from the methods on elliptic curves given in the duplicate answer. – Dietrich Burde Apr 02 '25 at 12:39
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    @DietrichBurde thanks Mr. Dietrich, I found Kieth Conrad's work on Mordell Curve helpful in general. – K41 Apr 02 '25 at 12:53
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    Yes, I also find this a very nice article. By the way, his name is Keith. – Dietrich Burde Apr 02 '25 at 15:54

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