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Prove that \(2005^2\) can be written as the sum of \(2\) perfect squares in at least \(4\) ways.

Note by Cody Johnson 3 years, 3 months ago

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2 \times 3

2^{34}

a_{i-1}

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Solution (not mine): Notice that \(2005=|(1\pm2i)(1\pm20i)|\), so \(2005^2=|(1\pm2i)(1\pm20i)(1\pm2i)(1\pm20i)|\). Choose the \(\pm\)'s to get

\[2005^2=1037^2+1716^2=1203^2+1604^2=1995^2+200^2=1357^2+1476^2\]

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\(This\ is\ just\ one\ way.\) \(We\ are\ basically\ looking\ for\ solutions\ for\ 2005^{2}=a^{2}+b^{2}.\) \(Thus\ we\ are\ looking\ for\ a\ pythagorean\ triplet\ (a,b,2005).\) \(Now\ let\ us\ keep\ that\ aside.\) \(We\ know\ that\ (3,4,5)\ is\ a\ pythagorean\ triplet.\) \(\Longrightarrow\)\(3x,4x,5x\ is\ also\ a\ pythagorean\ triplet.\) \(now,because\ 2005=5*401,we\ take\ x\ to\ be\ 401.\) \(\Longrightarrow\)\((3*401,4*401,5*401)\ is\ also\ a\ pythagorean\ triplet.\) \(\Longrightarrow\)\(1203^{2}+1604^{2}=2005^{2}\)

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`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestSolution (not mine): Notice that \(2005=|(1\pm2i)(1\pm20i)|\), so \(2005^2=|(1\pm2i)(1\pm20i)(1\pm2i)(1\pm20i)|\). Choose the \(\pm\)'s to get

\[2005^2=1037^2+1716^2=1203^2+1604^2=1995^2+200^2=1357^2+1476^2\]

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\(This\ is\ just\ one\ way.\) \(We\ are\ basically\ looking\ for\ solutions\ for\ 2005^{2}=a^{2}+b^{2}.\) \(Thus\ we\ are\ looking\ for\ a\ pythagorean\ triplet\ (a,b,2005).\) \(Now\ let\ us\ keep\ that\ aside.\) \(We\ know\ that\ (3,4,5)\ is\ a\ pythagorean\ triplet.\) \(\Longrightarrow\)\(3x,4x,5x\ is\ also\ a\ pythagorean\ triplet.\) \(now,because\ 2005=5*401,we\ take\ x\ to\ be\ 401.\) \(\Longrightarrow\)\((3*401,4*401,5*401)\ is\ also\ a\ pythagorean\ triplet.\) \(\Longrightarrow\)\(1203^{2}+1604^{2}=2005^{2}\)

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