# Spotting square numbers

Discussion in 'Mathematics' started by D Franklin, Nov 2, 2011.

1. ### D Franklin

This sounds like stating the obvious, but a number is square iff its square root is an integer.

So if you have a calculator, use the square root function.

if you don't have a calculator, you may be able to use inequalities. e.g. 1000 isn't square because 31^2 = 961 and 32^2 = 1024.

You can also rule out some possibilities: a square must end in 0, 1, 4, 5, 6 or 9.

Hmmm. I don't think there'll be one easy way, but there are probably some things you can look out for.

For example square numbers can never end in 2, 3, 7 or 8 for example, since the integers 0 - 9 all square to something ending in 0, 1, 4, 5, 6 or 9.

3. ### pec01

A non calc method would be to expresss the number in prime factors in index form, if all powers are even then square number.

4. ### etotheip

This is interesting. Thank you and a non- calculator method was something I was looking for

5. ### D Franklin

Note that there is only a fairly small window (numbers between about 1000 and 10000) where it's easier to factor them than it is to calculate their square root manually. If you want to tell if an 6 digit number is a perfect square, I'd say it will (usually) be a lot easier to find the square root.

6. ### frustumStar commenter

A bit of estimation can help, especially combined with last digits.
For instance, suppose you're looking at 641225.
That's not much more than 640000, so you're looking at 8xx. And it ends in 5, so if it's a square, it would have to end in 5. Check 805, and you're done.

7. ### Joslan

Have you tried dividing the number by 8? It is a fact that if a number is divided by 8 and the remainder is 0, 1 or 4 then the number is a square number.

8. ### AMathsTutor

No it isn't.
n^2 mod 8 = 0 1 or 4.
If it's square you get those remainders but if you get those remainders it isn't necessarily square.
Take 24 as an example.

Factors pairs. A square number has an odd number of factors.
If you pair up the factors of a square number you should the square root left over.
Does this help?
or did you want something for 4 digit numbers, 5 digit and 6 digit numbers specifically?
You can use the idea that square numbers are generated as a series by adding together a sequence of consectutive odd numbers.
So conversely the difference between consecutive square numbers are conscutive odd numbers.
Starting with the first 4 digit square number
(32)^2 = 1024
Diff 65
(33)^2 = 1089
Diff 67
(34)^2 = 1156
Diff 69
(35)^2 = 1225
Notice that the next difference [ d] is connected the number being squared [r]
d = 2r+1
So to find without a calculator [i.e a pencil and paper method] if a number is square then find an easy square near the value and work backwards [or forwards]
for example
test 2789
to get 2500+101 = 2601 for (51)^2
then add the next odd number = +103
to get 2601+103 = 2704
the next odd number is 105
2704 + 105 = 2809
So 2789 is not square

This "method" does help you to quickly sort between square and non-square integers.
I hope this helps.

10. ### AMathsTutor

Isn't it easier to simply square them, in the second case using (800-2)^2=640004-3200=636804?

You might notice that 53^2 > 2789 so 2789 is not square.

You might notice the orginal poster put
IMO AMT you are missing the point.

12. ### AMathsTutor

hmmm, I think you might be right.

I just did 7136849.

I had to square 2670 and 2673. Your way I squared 2670 and added 5341 and 5343 which I admit is less effort.

What did the OP mean by 1000 E though?