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Math Help - Showing that the sum of three squares is not closed

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    Showing that the sum of three squares is not closed

    Legendre: The numbers 3 and 21 can easily be represented as the sum of three squares: 3=1+1+1, 21=16+4+1, but *the product 3*21=63 cannot be so represented since 63 is of the form (8n+7)*.

    Can someone explain the starred statement?
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    Quote Originally Posted by Josh146 View Post
    Legendre: The numbers 3 and 21 can easily be represented as the sum of three squares: 3=1+1+1, 21=16+4+1, but *the product 3*21=63 cannot be so represented since 63 is of the form (8n+7)*.

    Can someone explain the starred statement?
    Integer as Sum of Three Squares - ProofWiki: no number of the form 8n+7 can be expressed as a sum of three squares.
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