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MathGroup Archive 2006

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Re: Number-Theory :: All-Digit Perfect Squares

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64286] Re: Number-Theory :: All-Digit Perfect Squares
  • From: Bill Rowe <readnewsciv at earthlink.net>
  • Date: Thu, 9 Feb 2006 02:45:06 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

On 2/8/06 at 3:54 AM, bdsatish at gmail.com (bd satish) wrote:

>I set out to write a code that generates nine-digit
>perfect square numbers , with each of the digits 1,2,3,...9
>occuring only once in a given number. An example is 139854276  =
>11826^2 . Obviuously , all such numbers must lie in the interval
>[123456789 , 987654321] . Since 11111^2 < 123456789 and 31427^2 >
>987654321 ,  all the square numbers must have their square-roots in
>the interval (11111,31427) .

<code snipped>

The square roots of the desired numbers can be generated in one line as:

In[8]:=
s = Cases[Range[11111, 31427], 
   _?(Length@Complement[IntegerDigits[#^2],{0}] == 9&)]

Out[8]=
{11826, 12363, 12543, 14676, 
  15681, 15963, 18072, 19023, 
  19377, 19569, 19629, 20316, 
  22887, 23019, 23178, 23439, 
  24237, 24276, 24441, 24807, 
  25059, 25572, 25941, 26409, 
  26733, 27129, 27273, 29034, 
  29106, 30384}

and the desired numbers can be obtained by simply squaring s

Or if you prefer having the 9 digit numbers in one line rather than the square roots simply change where the squaring takes place, i.e.,

Cases[Range[11111, 31427]^2, 
   _?(Length@Complement[IntegerDigits[#],{0}] == 9&)]
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