Number Theory #4

Let $n$ be a positive integer such that $2n+1$ and $3n+1$ are both squares. Prove that $40|n$ .

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

It suffices to prove $5\mid n,8\mid n$ .

Given that $2n+1=x^2, 3n+1=y^2$ , If we add the two equations we get: $5n+2=x^2+y^2\Rightarrow x^2+y^2\equiv 2\pmod 5$ , since the quadratic residues of mod $5$ are $0,1,4$ , we can check that only $x^2\equiv y^2\equiv 1$ works, therefore $2n\equiv 1-1\equiv 0\pmod 5\Rightarrow 5\mid n$

The same analysis can be done for $8$ , but we will multiply the second equation by $2$ first and then add: $8n+3=x^2+2y^2\Rightarrow x^2+2y^2\equiv 3\pmod 8$ , since the quadratic residues of mod $8$ are $0,1,4$ , we can check that only $x^2\equiv y^2\equiv 1$ works again, which means $3n\equiv 1-1\equiv 0\pmod 8\Rightarrow 8\mod n$ and we are done.

Xuming Liang - 6 years, 11 months ago

Great!

Victor Loh - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$