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Perfect Squares: n+125 and n+201


Date: 01/21/2002 at 06:43:52
From: Lisa
Subject: Perfect squares

Find the smallest positive integer n so that n+125 and n+201 are both 
perfect squares.


Date: 01/21/2002 at 11:12:18
From: Doctor Paul
Subject: Re: Perfect squares

The answer is 199.  

Here's how to do it:

Can you make a list of perfect squares?

  1
  4
  9
 16
 25
 36
 49
 64
 81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625   

What values of n make n+125 a perfect square?

To solve the above question, you need to subtract 125 from every 
number on the list of perfect squares that is greater than 125.

This gives:

 19
 44
 71
100
131
164
199
236
275
316
359
404
451
500

Now, what values of n make n+201 a perfect square?

To solve the above question, you need to subtract 201 from every 
number on the list of perfect squares that is greater than 201.

 24
 55
 88
123
160
199
240
283
328
375
424

To complete the problem, we're looking for the smallest number on both 
lists - i.e., the smallest number that makes both n+125 and n+201 a 
perfect square. Clearly, the answer is 199 since 199+125 = 324 = 18^2 
and 199+201 = 400 = 20^2.

I wanted to know what the next number was that makes n+125 and n+199 a 
perfect square so I wrote a little program and asked my computer to 
find it.  I gave up at ten million...

I hope this helps.  Please write back if you'd like to talk about this 
more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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