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• In a message dated 4/3/2001 7:01:39 AM Pacific Daylight Time, ... Observed heuristically, if you sum the digits of a square recursively to a single digit, the
Message 1 of 7 , Apr 3, 2001
In a message dated 4/3/2001 7:01:39 AM Pacific Daylight Time,
Mnemonix@... writes:

> Does anybody know of a quick test to see whether a number squareroots to
> give a whole number? sqrt() is too computationally "heavy" and I don't need
> the actual squareroot - I just need the check to see if the number
> squareroots to give a whole number.
>

Observed heuristically, if you sum the digits of a square recursively to a
single digit, the squares will sum to 1, 4, 7, or 9 only. For example 5329-->
19-->10-->1.

Also the square sums have roots with specific sums:

1 squares have roots that sum to 1 or 8
4 squares have roots that sum to 2 or 7
7 squares have roots that sum to 4 or 5
9 squares have roots that sum to 3, 6 or 9

I think this is equivalent to one of the methods shown earlier and only rules
out some numbers.

Steve Wagler

[Non-text portions of this message have been removed]
• ... Indeed. It s the quadratic residue test with modulus 9. It s subsumed in the mod 63 test I gave and it s equivalent in the other proposals. Paul
Message 2 of 7 , Apr 4, 2001
> Observed heuristically, if you sum the digits of a square
> recursively to a
> single digit, the squares will sum to 1, 4, 7, or 9 only. For
> example 5329-->
> 19-->10-->1.
>
> Also the square sums have roots with specific sums:
>
> 1 squares have roots that sum to 1 or 8
> 4 squares have roots that sum to 2 or 7
> 7 squares have roots that sum to 4 or 5
> 9 squares have roots that sum to 3, 6 or 9
>
> I think this is equivalent to one of the methods shown
> earlier and only rules out some numbers.

Indeed. It's the quadratic residue test with modulus 9. It's subsumed
in the mod 63 test I gave and it's equivalent in the other proposals.

Paul
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