Forbidden numbers: Difference between revisions

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Others require at least three squares. <code>'''6 == 22 + 12 + 12'''</code>

Finally, some, (the focus of this task) require the sum at least four squares. <code>'''7 == 22 + 12 + 12 + 12'''</code>. There is no way to reach 7 ~~other~~ ~~than~~ ~~summing~~ four squares.

Finally, some, (the focus of this task) require the sum at least four squares. <code>'''7 == 22 + 12 + 12 + 12'''</code>. There is no way to reach 7 summing fewer than four squares.

These numbers show up in crystallography; x-ray diffraction patterns of cubic crystals depend on a length squared plus a width squared plus a height squared. Some configurations that require at least four squares are impossible to index and are colloquially known as '''forbidden numbers'''.

Note that some numbers ''can'' be made from the sum of four squares: <code>'''16 == 22 + 22 + 22 + 22'''</code>, but since it can also be formed from ~~less~~ than four, <code>'''16 == 42'''</code>, it does not count as a forbidden number.

Note that some numbers ''can'' be made from the sum of four squares: <code>'''16 == 22 + 22 + 22 + 22'''</code>, but since it can also be formed from fewer than four, <code>'''16 == 42'''</code>, it does not count as a forbidden number.

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 Others require at least three squares. <code>'''6 == 2<sup>2</sup> + 1<sup>2</sup> + 1<sup>2</sup>'''</code>
-Finally, some, (the focus of this task) require the sum at least four squares. <code>'''7 == 2<sup>2</sup> + 1<sup>2</sup> + 1<sup>2</sup> + 1<sup>2</sup>'''</code>. There is no way to reach 7 other than summing four squares.
+Finally, some, (the focus of this task) require the sum at least four squares. <code>'''7 == 2<sup>2</sup> + 1<sup>2</sup> + 1<sup>2</sup> + 1<sup>2</sup>'''</code>. There is no way to reach 7 summing fewer than four squares.
 These numbers show up in crystallography; x-ray diffraction patterns of cubic crystals depend on a length squared plus a width squared plus a height squared. Some configurations that require at least four squares are impossible to index and are colloquially known as '''forbidden numbers'''.
-Note that some numbers ''can'' be made from the sum of four squares: <code>'''16 == 2<sup>2</sup> + 2<sup>2</sup> + 2<sup>2</sup> + 2<sup>2</sup>'''</code>, but since it can also be formed from less than four, <code>'''16 == 4<sup>2</sup>'''</code>, it does not count as a forbidden number.
+Note that some numbers ''can'' be made from the sum of four squares: <code>'''16 == 2<sup>2</sup> + 2<sup>2</sup> + 2<sup>2</sup> + 2<sup>2</sup>'''</code>, but since it can also be formed from fewer than four, <code>'''16 == 4<sup>2</sup>'''</code>, it does not count as a forbidden number.