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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

Consider an integer x. If we add 30, then the result is a perfect square. If we subtract 30, the result is also a perfect square. How many such integers are there?"

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Hi anna

Did you try setting up the equations? Let us know what you have tried.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,394

Hi all;

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Hi bobby

That is not the only one.There is one more solution.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,847

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,394

That is not the only one.There is one more solution.

That is true, but only one more?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Yes only one. Look at phro's answer.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,394

How about one past where he looked?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

What?

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,394

He must have searched up to some number. That was the limit of his search. Can you provide a reason why there is not a number passed his search?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Yes I can.

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,394

When you have the time please post your proof.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,847

Hi Bobby,

I only searched as far as shown in my image, because at that point:

The numbers that are squared (I don't know what they're called) to produce the perfect squares must differ by at least 1.

*EDIT: The column E heading should be "If Col D = integer, print B + 30".*

*Last edited by phrontister (2012-04-11 21:20:13)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

And I did it non-experimentally!

Here lies the reader who will never open this book. He is forever dead.

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**wintersolstice****Real Member**- Registered: 2009-06-06
- Posts: 114

anonimnystefy wrote:

And I did it non-experimentally!

I did it diferent again:D

*Last edited by wintersolstice (2012-04-11 21:44:11)*

Why did the chicken cross the Mobius Band?

To get to the other ...um...!!!

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,847

Hi anonimnystefy,

Yes, I like that proof. Good reasoning, and with a process of elimination at the end that leaves only those two possibilities.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

Here is my solution: It is based to the fact that each perfect square N^2 is the sum of the first N odd numbers (5^2 = 25 = 1+3+5+7+9).

Thus the difference of any 2 perfect squares should equal to the sum of consecutive odd numbers (and this should equal 60).

Starting from 1, we write down the sums of the odd numbers:

1+3+5+7+9+11+13 = 49 while 1+3+5+7+9+11+13+15 = 64. Thus we cannot make 60 starting from 1.

We do the same with 3: 3+5+7+9+11+13=48 while 3+5+7+9+11+13+15 = 63. Not possible.

Starting from 5:

5+7+9+11+13+15=60 Here we are.

So, one perfect square is 4 and the next is 64, their difference being 60, so the first number we are looking for is 34 (34-30 = 4 and 34+30 = 64, both of them perfect squares).

Similarly, we find that the only other sum of consecutive odd numbers equaling 60 is 29+31.

Therefore one perfect square is 1+3+5+...+27=196 (14^2) and the next is 1+3+5+...+27+29+31=256 (16^2) and the second number we are asking for is 226 (226-30 = 196, 226+30 = 256).

There is no other series of successive odd numbers equaling 60, so these two are the only numbers with this property.

Obviously this is not a proper "proof"; it is more based on a "guess and try" method, but it works!

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**wintersolstice****Real Member**- Registered: 2009-06-06
- Posts: 114

anna_gg wrote:

Here is my solution: It is based to the fact that each perfect square N^2 is the sum of the first N odd numbers (5^2 = 25 = 1+3+5+7+9).

Thus the difference of any 2 perfect squares should equal to the sum of consecutive odd numbers (and this should equal 60).

Starting from 1, we write down the sums of the odd numbers:

1+3+5+7+9+11+13 = 49 while 1+3+5+7+9+11+13+15 = 64. Thus we cannot make 60 starting from 1.

We do the same with 3: 3+5+7+9+11+13=48 while 3+5+7+9+11+13+15 = 63. Not possible.

Starting from 5:

5+7+9+11+13+15=60 Here we are.

So, one perfect square is 4 and the next is 64, their difference being 60, so the first number we are looking for is 34 (34-30 = 4 and 34+30 = 64, both of them perfect squares).

Similarly, we find that the only other sum of consecutive odd numbers equaling 60 is 29+31.

Therefore one perfect square is 1+3+5+...+27=196 (14^2) and the next is 1+3+5+...+27+29+31=256 (16^2) and the second number we are asking for is 226 (226-30 = 196, 226+30 = 256).

There is no other series of successive odd numbers equaling 60, so these two are the only numbers with this property.

Obviously this is not a proper "proof"; it is more based on a "guess and try" method, but it works!

Have you seen my proof? It's very similar in that it's based on consecutive odd numbers.

Why did the chicken cross the Mobius Band?

To get to the other ...um...!!!

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