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#1 2015-04-25 05:19:17
Powers of Two and Squares
Find the first five positive integers x and n such that x^2 + 615 == 2^n or prove that they do not exist
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#2 2015-04-25 16:19:53
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
Hi;
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#3 2015-04-25 16:28:32
Re: Powers of Two and Squares
Great! That is the solution from @trevorarashiro
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#4 2015-04-25 16:38:32
- Olinguito
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Re: Powers of Two and Squares
Last edited by Olinguito (2015-04-25 16:52:00)
Bassaricyon neblina
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#5 2015-04-25 16:56:52
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
Hi Olinguito;
Your example changes the problem considerably. I do not think you have to worry about when n is odd. There seems to be a fairly easy way to show this, which I left out.
I did not include it in my proof because I expect that Mr. Chattopadhyay will soon ask the same question you did and then I would.
Great! That is the solution from @trevorarashiro
Also, it is incorrect to say that my demonstration is similar to his. Mine is a computational attack on the problem that I wanted to show you. I expected you to ask many questions about that answer.
In addition, why post the addition when in all likelihood you and/or Bob will post something much better...
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#6 2015-04-25 18:10:06
Re: Powers of Two and Squares
What is the addition?
If n were odd, then the last digit of the RHS would be 2 or 8 which is not possible.
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#7 2015-04-25 18:13:35
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
Another proof that n can not be odd.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#8 2015-04-25 18:16:24
Re: Powers of Two and Squares
>I expected you to ask many questions about that answer.
Yep, I love asking questions.
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#9 2015-04-25 18:23:28
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
Did you try the code?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#10 2015-04-25 18:54:11
Re: Powers of Two and Squares
No. I wrote my own code last night
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#11 2015-04-25 22:07:01
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
What does it look like?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#12 2015-04-25 23:19:40
Re: Powers of Two and Squares
Running over around 1000 natural numbers to check if there is a solution
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.
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#13 2015-04-26 06:53:06
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
That is very good.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#14 2015-04-26 21:00:48
- phrontister
- Real Member
- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,895
Re: Powers of Two and Squares
Hi;
For n<10000 I got just one solution:
That took 18 seconds, so I didn't try any greater n.
Last edited by phrontister (2015-04-26 22:32:56)
"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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#15 2015-04-27 04:13:05
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
Hi;
That is the only one there is.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#16 2015-04-27 11:57:43
- phrontister
- Real Member
- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,895
Re: Powers of Two and Squares
Hi Bobby,
Yes, I took it that your code - which I ran in M - shows there is only one solution. However, I couldn't understand it and so I thought I'd try some code myself, and although I found the sole solution I wasn't able to prove that it was the only one.
Last edited by phrontister (2015-04-27 12:00:03)
"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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#17 2015-04-27 12:37:17
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
It takes some math and some code to do that, M can do both.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#18 2015-04-27 12:46:35
- phrontister
- Real Member
- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,895
Re: Powers of Two and Squares
Could you give me a clue about the maths?
"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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#19 2015-04-27 15:45:22
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
Hi;
Try this piece of code:
Factor[2^(2 n) - x^2]In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#20 2015-04-27 16:15:04
- phrontister
- Real Member
- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,895
Re: Powers of Two and Squares
Yes, I'd already seen that you used the 'difference between two squares' idea in post #2, but I didn't (and still don't) know what to do with that info.
Also, I don't understand how the only area in which a solution might exist is within the range you used, which involved just the four possible factor combinations of 615.
Btw, is this the sort of thing that individ might have a set of solution equations for? x^2 + 615 = 2^n is a diophantine equation, isn't it?
"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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#21 2015-04-27 16:41:30
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
We will do it step by step and then it will get clear.
When you set the equation up as 2^n - x^2 = 615 you now factor 615
Divisors[615]In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#22 2015-04-27 17:24:56
- phrontister
- Real Member
- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,895
Re: Powers of Two and Squares
Ok...done.
Btw, Excel's Solver gave the sole solution as its answer, but as it can only display one result even if there are several, that still leaves the door open for more solutions.
"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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#23 2015-04-27 17:30:39
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
You should have {1, 3, 5, 15, 41, 123, 205, 615}.
Now pair them off two at a time to make a product of 615. For instance 1 x 615 = 615.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#24 2015-04-27 17:32:07
- phrontister
- Real Member
- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,895
Re: Powers of Two and Squares
Done...same result as your post #2.
"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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#25 2015-04-27 17:40:25
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Powers of Two and Squares
You got 4 pairs of numbers?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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