Solutions to Diophantine Equation 16x^2+32x+20 = y^2+y
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Theorem
The indeterminate Diophantine equation:
- $16x^2 + 32x + 20 = y^2 + y$
has exactly $4$ solutions:
- $\tuple {0, 4}, \tuple {-2, 4}, \tuple {0, -5}, \tuple {-2, -5}$
Proof
\(\ds 16 x^2 + 32 x + 20\) | \(=\) | \(\ds y^2 + y\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 16 x^2 + 32 x + 16 + 4\) | \(=\) | \(\ds \) | |||||||||||
\(\ds 16 \paren {x^2 + 2 x + 1} + 4\) | \(=\) | \(\ds \) | ||||||||||||
\(\ds 16 \paren {x + 1}^2 + 4\) | \(=\) | \(\ds y^2 + y\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 64 \paren {x + 1}^2 + 16\) | \(=\) | \(\ds 4 y^2 + 4 y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {8 x + 8}^2 + 17\) | \(=\) | \(\ds 4 y^2 + 4 y + 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {8 x + 8}^2 + 17\) | \(=\) | \(\ds \paren {2 y + 1}^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 17\) | \(=\) | \(\ds \paren {2 y + 1}^2 - \paren {8 x + 8}^2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 y + 1 - 8 x - 8} \paren {2 y + 1 + 8 x + 8}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 y - 8 x - 7} \paren {2 y + 8 x + 9}\) |
$17$ is prime and therefore the solution of only two sets of integer products:
\(\ds 17\) | \(=\) | \(\ds 1 \times 17\) | ||||||||||||
\(\ds 17\) | \(=\) | \(\ds -1 \times -17\) |
This leaves us with four systems of equations with four solutions:
\(\ds 1\) | \(=\) | \(\ds 2 y - 8 x - 7\) | ||||||||||||
\(\ds 17\) | \(=\) | \(\ds 2 y + 8 x + 9\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 1 + 17\) | \(=\) | \(\ds 2y - 8x + 9 + 2y + 8x - 7\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 18\) | \(=\) | \(\ds 4y + 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4\) | \(=\) | \(\ds y\) | |||||||||||
\(\ds 1\) | \(=\) | \(\ds 2 \paren 4 - 8 x - 7\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(=\) | \(\ds x\) |
Hence the solution:
- $\tuple {0, 4}$
\(\ds 17\) | \(=\) | \(\ds 2 y - 8 x - 7\) | ||||||||||||
\(\ds 1\) | \(=\) | \(\ds 2 y + 8 x + 9\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 17 + 1\) | \(=\) | \(\ds 2 y - 8 x - 7 + 2 y + 8 x + 9\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 18\) | \(=\) | \(\ds 4 y + 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4\) | \(=\) | \(\ds y\) | |||||||||||
\(\ds 1\) | \(=\) | \(\ds 2 \tuple 4 + 8 x + 9\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds -2\) | \(=\) | \(\ds x\) |
Hence the solution:
- $\tuple {-2, 4}$
\(\ds -17\) | \(=\) | \(\ds 2 y - 8 x - 7\) | ||||||||||||
\(\ds -1\) | \(=\) | \(\ds 2 y + 8 x + 9\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds -1 - 17\) | \(=\) | \(\ds 2 y - 8 x + 9 + 2 y + 8 x - 7\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -18\) | \(=\) | \(\ds 4 y + 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -5\) | \(=\) | \(\ds y\) | |||||||||||
\(\ds -1\) | \(=\) | \(\ds -2 \paren {-5} - 8 x - 7\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(=\) | \(\ds x\) |
Hence the solution:
- $\tuple {0, -5}$
\(\ds -1\) | \(=\) | \(\ds 2 y - 8 x - 7\) | ||||||||||||
\(\ds -17\) | \(=\) | \(\ds 2 y + 8 x + 9\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds -1 - 17\) | \(=\) | \(\ds 2 y - 8 x + 9 + 2 y + 8 x - 7\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -18\) | \(=\) | \(\ds 4 y + 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -5\) | \(=\) | \(\ds y\) | |||||||||||
\(\ds -1\) | \(=\) | \(\ds 2 \paren {-5} - 8 x - 7\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds -2\) | \(=\) | \(\ds x\) |
Hence the solution:
- $\tuple {-2, -5}$
$\blacksquare$
Also see
Sources
- Kieren-MacMillan (https://math.stackexchange.com/users/93271/Kieren-MacMillan), How to solve inhomogeneous quadratic forms in integers, URL (version: 2019-07-16): https://math.stackexchange.com/q/9269