An interesting solution! I was hoping for something more elementary though (one where people who don't have experience in number theory could potentially solve through brute force).(Original post by Renzhi10122)
Solution 553
We construct a sequence that works. Start with then let be some large number such that we may have , i.e . Now let be some even larger number compared to such that we may have , and such a number exists by CRT. Hence, we keep repeating this, alternating the sequence between large number and the sequence , missing out a part of the sequence if that number has already been used as one of our large numbers, and hence, we have constructed a sequence that works.

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Renzhi10122
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Blazy's problem reminded me of this problem
Problem 554 *
Let be a sequence of integers with infinitely many positive and infinitely many negative terms. Suppose that for every positive integer , the numbers leave distinct remainders upon division by .
Prove that every integer occurs in the sequence exactly once.Last edited by Renzhi10122; 24122015 at 00:38. 
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Johann von Gauss
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Last edited by Johann von Gauss; 24122015 at 01:03. 
Renzhi10122
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(Original post by Johann von Gauss)
What is the domain of n?
Are you stating
Else if we can take any real n, then
Assume holds for some integer q
Let 
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 24122015 01:01
(Original post by Renzhi10122)
n is integer, seeing as it doesnt work in the reals nor the rationals. Also, what on earth does your first line mean?
i.e. there exist no nonnatural real q for which his statement is true
Statements within statements is a key feature of my work
I find it more logically intuitiveLast edited by Johann von Gauss; 24122015 at 01:04. 
Lord of the Flies
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 24122015 01:07
(Original post by Johann von Gauss)
... 
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 24122015 01:15
(Original post by Johann von Gauss)
(There exists a real q such that for all natural n, we have n^q is a natural number) implies q is a natural number
i.e. there exist no nonnatural real q for which his statement is true
Statements within statements is a key feature of my work
I find it more logically intuitive 
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 24122015 01:23
(Original post by Lord of the Flies)
Oh come on obviously . And you could've said all that in a single sentence without the silly syntax.
And the syntax isn't silly, its unambiguous, and quick to read
(Original post by Renzhi10122)
Ah ok, im just not a fan of the big P, Q and T business.Last edited by Johann von Gauss; 25122015 at 14:48. 
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(Original post by Johann von Gauss)
It saves ink
Btw what level is your problem? I'm a little afraid to give it a serious attempt, if its some IMO level problem
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 24122015 01:47
(Original post by Johann von Gauss)
What is the domain of n?
Are you stating
Else if we can take any real n, then
Assume
Let
You can drop the s.t. also, to make it more compact and avoid those big bold letters for propositions as they're used for sets.
I like it though.Last edited by EricPiphany; 24122015 at 02:09. 
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 24122015 01:53
(Original post by Johann von Gauss)
...
In the case of 555 it is crystal clear what the domain should be, since otherwise the question wouldn't be worth asking, as you've needlessly pointed out.
And terribly sorry, but unless you're writing a paper on predicate logic it most definitely is silly, unnecessary, and unimpressive. Open any article by any mathematician ever and you can be sure that any statements made won't be written like that  for good reason.
Not going to elaborate if you bother responding because I have better things to do than convince a sixthformer who probably discovered quantifiers a week ago that their syntax would make any undergrad chuckle. 
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 24122015 08:37
^
Fair enough, but would someone else be kind enough to explain why this kind of syntax isn't used outside logic?
I had thought that University level maths and above were very rigorous in their logical connectives
Doesn't using words (which are ambiguous) go against this, especially when not all mathematicians speak English well?
From a quick search, I have come across https://en.wikibooks.org/wiki/Mathem...cal_Connectors which seems to encourage this sort of syntax:
Last edited by Johann von Gauss; 24122015 at 08:58. 
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 24122015 10:23
(Original post by Johann von Gauss)
^
Fair enough, but would someone else be kind enough to explain why this kind of syntax isn't used outside logic?
I had thought that University level maths and above were very rigorous in their logical connectives
Doesn't using words (which are ambiguous) go against this, especially when not all mathematicians speak English well?
From a quick search, I have come across https://en.wikibooks.org/wiki/Mathem...cal_Connectors which seems to encourage this sort of syntax:
Mathematics is inherently an art and constricting yourself to expressing it through a particular sybtax destroys that inherent beauty, imo.
Sentences and English do a much better job in coveying mathematical ideas clearly, plus a nice diagram or two if required.
Do excuse my spelling, I'm typing on my phone and that's never good. 
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 24122015 10:44
(Original post by Johann von Gauss)
^
Fair enough, but would someone else be kind enough to explain why this kind of syntax isn't used outside logic?
(b) It is just as easy to make mistakes in using complex logical symbolism as it is in expressing the idea in English. In fact, easier to make such mistakes.
I had thought that University level maths and above were very rigorous in their logical connectives
Doesn't using words (which are ambiguous) go against this, especially when not all mathematicians speak English well? 
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 24122015 11:18
(Original post by Gregorius)
...
Personally, I usually find translating problems into logical syntax lets me see what is required easier  but this is clearly not a view shared by most people  so I shall cease and desist EDIT: in using it for anything other than organising my own thoughts
(Original post by Zacken)
Sentences and English do a much better job in coveying mathematical ideas clearly, plus a nice diagram or two if required.
I guess this depends on how you visualise the problems, and your's seems to be the majority view. But I have to say I find sentences ugly.Last edited by Johann von Gauss; 24122015 at 12:02. 
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(Original post by Johann von Gauss)
Thank you for your helpful response
Personally, I usually find translating problems into logical syntax lets me see what is required easier  but this is clearly not a view shared by most people  so I shall cease and desist. 
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(Original post by qwertzuiop)
x=2 as squaring the LHS gives + sqrt(3)  sqrt(3) [=0], and 2+2 which equals 4 which = RHS. Simples.
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