Re: Fast way of checking for perfect squares?

• To: mathgroup at smc.vnet.net
• Subject: [mg83542] Re: Fast way of checking for perfect squares?
• From: dh <dh at metrohm.ch>
• Date: Thu, 22 Nov 2007 04:56:31 -0500 (EST)
• References: <fhu6m6\$6pd\$1@smc.vnet.net>

```
Hi Mike,

here is still another pretty fast method:

Calculate all possible squares and take the intersection with your data:

E.g.:

data=Table[RandomInteger[n=10000000000],{100000}];

t=Range[100000]^2;//Timing

Intersection[t,data]//Timing

this produces:

{0.078,Null}

{0.266,{2876498689}}

this takes a time of approx. 0.3 sec.

hope this helöps, Daniel

> Hi

>

> Lets say I have a lot of large integers and I want to check to see

> which ones are perfect squares - eg

>

> data = Table[RandomInteger[10000000000], {100000}];

>

> I create a function that takes the square root of each one and checks

> to see if the result is an integer

>

> slowSquareQ := IntegerQ[Sqrt[#1]] &

>

> This works fine:

>

> Select[data, slowSquareQ] // Timing

>

> {11.39, {6292614276, 2077627561}}

>

> but I am wondering if I can make it faster as my actual application

> has a LOT more numbers to test than this.  This was a question posed a

> few years ago in this very newsgroup and the suggestion was to use a

> test of the form MoebiusMu[#]==0.

>

> Well the MoeboisMu function is new to me so I experimented a little

> and sure enough when you apply the MoebiusMu function to a perfect

> square number the result is zero.

>

> MoebiusMu[4] = 0

>

> The problem is that lots of other numbers have this property as well -

> eg

>

> MoebiusMu[8] =0

>

> despite this minor problem I determined that applying the test

> MoebiusMu[#]==0 on a list of integers is fater than my slowSquareQ:

>

> mobSquareQ := If[MoebiusMu[#1] == 0, True, False] &

> Select[data, mobSquareQ]; // Timing

>

> gives a result of 8.156 seconds compared to 11.39 for slowSquareQ.  On

> my test data I had around 39,000 integers that passed this test which

> is a shame but at least I have eliminated the other 61,000 or so.  So

> I thought that maybe I can use the faster MoebiusMu test as a filter:

>

> SquareQ := If[MoebiusMu[#1] == 0, slowSquareQ[#1], False] &

> Select[data, SquareQ] // Timing

> {11.312, {6292614276, 2077627561}}

>

> So after all that I have saved just a few tenths of a second.  Not

> very impressive.  Can anyone out there do any better?

> Please forgive me if I have made any stupid mistakes but I have a cold

> at the moment.

>

> Best regards,

> Mike

>

>

>

>

>

>

>

>

```

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