Problem 195*/**
Prove that there exists infinitely many arithmetic progressions of 3 distinct perfect squares.
Problem 196**/***
Prove that there exists no arithmetic progressions of 4 distinct perfect squares.
Problem 197*/**
For a nonnegative integer n, is ever prime? Prove your assertions.
Problem 198**
Let the real numbers a,b,c,d satisfy the relations and .
Prove that
Problem 199***
Prove that
Hint:Spoiler:ShowConvolution
Problem 200*/***
Evaluate

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 01062013 15:44
Last edited by Jkn; 01062013 at 16:12. 
Felix Felicis
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Last edited by Felix Felicis; 02062013 at 03:24. Reason: tex 
ukdragon37
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(Original post by Implication)
Isn't that because mathematical induction is actually deduction haha?
(Original post by Implication)
Interestingly, I'm not sure if that is a solution the Problem of Induction: you still need to find a way to justify your confidence level, do you not? And how are you going to do that, if not by reasoning inductively in the first place? And for this you will need another confidence interval. And so on, ad infinitum.
(Original post by Implication)
There's a problem very closely linked to the problem of induction that I've always found fascinating and extremely frustrating. Again, I think it is a pretty old di/tri/multilemma, but it's always fun to think about!
Any valid argument must rely on either a circular chain of reasons, an infinite chain of reasons or on reasons that are not themselves justified.
An argument cannot be justified by a circular chain of reasons.
(Original post by Implication)
An argument cannot be justified by an infinite chain of reasons.
(Original post by Implication)
An argument cannot be justified if it relies upon unjustified assumptions.
(Original post by Implication)
Thus, no valid argument can be justified.
Tl;dr: What you said appears to be a trilemma only because you are confining yourself to classic logic, and even then it's a false one since it ignores the fact that arguments are justified only relative to some axioms.Last edited by ukdragon37; 01062013 at 16:20. 
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 01062013 16:18
(Original post by Felix Felicis)
Solution 200
Why is this ***? I presume you want justification of the limit...
Spoiler:ShowBtw, typo in your first line 
metaltron
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 01062013 16:21
(Original post by Jkn)
Problem 195*/**
Prove that there exists infinitely many arithmetic progressions of 3 distinct perfect squares.
Problem 196**/***
Prove that there exists no arithmetic progressions of 4 distinct perfect squares.
Problem 197*/**
For a nonnegative integer n, is ever prime? Prove your assertions.
Problem 198**
Let the real numbers a,b,c,d satisfy the relations and .
Prove that
Problem 199***
Prove that
Hint:Spoiler:ShowConvolution
Problem 200***
Evaluate
Let
then is a prime. Since n is a nonnegative integer 5^{5^n} is also an integer. Hence, both brackets will be integers, so one of the brackets must be equal to 1 for the expression to be a prime. Since n is nonnegative, 5^n will be a positive integer, and it follows that 5^5^n is also a positive integer. Hence, all terms in the first bracket are positive and the sum must be > 1, so the first bracket cannot = 1. Leaving:
As above x will be a positive integer so not 0.
which is impossible since n would have to be negative infinity for the power to be 0, which is impossible firstly because n is a nonnegative integer. Hence, no such prime exists.Last edited by metaltron; 01062013 at 16:23. 
Felix Felicis
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 01062013 16:24
(Original post by Jkn)
Actually I found it in a list of problems to practice using the Gamma Function and results such as that obtained in 199 and so hadn't really stopped to consider how it might be approached outside of this context (the advanced method is still worth doing though if you're up for it!) Your solution is far better
Spoiler:ShowBtw, typo in your first line
Ehh, I don't know nearly enough about the Beta/ Gamma functions past their definitions to attempt questions with them I'll save further reading 'til my exams are over. 
Implication
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 01062013 17:43
(Original post by Zakee)
You say 'no valid argument' can be justified. Does that mean your argument is invalid as you have justified it?
So we need to look at where the problem might lie, and I think the answer is clear: the argument rests upon unjustified assumptions (axioms) as, I think, do most/all arguments. And so it boils down to the idea behind the problem in the first place: if an argument rests on assumptions that are not inferentially justified, then how can that argument ultimately be justified?
Now I think an axiom (one of the initial assumptions we don't justify) can be of two types; it can be a defining property of something (such as with most axioms in mathematics, I think), or it can be "selfevidently true". And I think, aside from tautologies that give no new information, the idea of anything about reality being "selfevidently" true is unrealistic. We could always be wrong about what is "selfevidently" true.
But hey, I'm just a young man trying to sound cleverer than he is I'm not a philosopher; nor do I have a mathematics degree. I could be entirely wrong 
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 01062013 17:58
(Original post by Implication)
No; not quite, I don't think. The argument I presented is clearly valid in that its premises entail its conclusion. Whether or not it is justified is another matter! If the conclusion is justified, then it is correct and the conclusion cannot be justified or we have a contradiction. So the conclusion cannot be justified. If the conclusion is not justified, then everything is fine i.e. there is no contradiction. So we must conclude that the conclusion is unjustified (though not necessarily false) as it must either be justified or unjustified and we have shown that it is not justified.
So we need to look at where the problem might lie, and I think the answer is clear: the argument rests upon unjustified assumptions (axioms) as, I think, do most/all arguments. And so it boils down to the idea behind the problem in the first place: if an argument rests on assumptions that are not inferentially justified, then how can that argument ultimately be justified?
Now I think an axiom (one of the initial assumptions we don't justify) can be of two types; it can be a defining property of something (such as with most axioms in mathematics, I think), or it can be "selfevidently true". And I think, aside from tautologies that give no new information, the idea of anything about reality being "selfevidently" true is unrealistic. We could always be wrong about what is "selfevidently" true.
But hey, I'm just a young man trying to sound cleverer than he is I'm not a philosopher; nor do I have a mathematics degree. I could be entirely wrong
There are probably inferences and statements which are so profoundly conspicuous, to justify them would be mere tautology.
For example, "This is", is a statement which is obvious and the only flaws viewed would have to be in the structure of language and how language is used. If we try and investigate the plights met in language, then well, I'd prefer to refrain from such as it'd probably drive me to insanity.
Also with respect to your last line: doesn't everyone try to do that? We all delude ourselves some way or another into thinking something that isn't true: that we're loved, or that we understand something or that we have some ineffable, majestic purpose in the Universe.Last edited by Zakee; 01062013 at 18:00. 
Implication
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 01062013 18:23
(Original post by ukdragon37)
No a confidence level constructed correctly is deductively justified, for example through interpretations of probability or statistical inference, which are sound albeit open to disputes of opinion.
Say we've measured the stress levels of managers and floor staff in a sample of shops and found that, in general, managers are more stressed than floor staff. We can do our stats to compute the probability that the difference in stress measurements between managers and floor staff is merely due to random chance. But, without any inductive inferences, we can't make any judgement about what did cause this difference: we can only decide whether it was likely to be due to random chance or a real variable difference. If it is likely to be due to a real variable, the stats still hasn't told anything about that variable beyond the fact that it isn't random chance.
With regard to the icebergs, wouldn't our confidence interval necessarily rely on assumptions and inductive inferences? We may well have measured 1 000 000 icebergs and done our stats to find that we can be 99.7% certain that all icebergs are cold, for example... But what if there is a different type of underground iceberg that forms under pressure  and at higher temperatures  that we didn't know about then and is subsequently discovered? The stats may well have told us that the probability was 99.7%, but, without sounding like an excitable philosophy freshman with an obsession with Descartes and his method of doubt, were we really in a position to make that call when we had no way of knowing what information and data we were missing regarding icebergs in the first place? Our statistical calculation didn't (and couldn't have) taken this into account, so how can we ever have really been so sure in the first place?
I really don't know that much about stats, as you can probably tell, so I'm struggling to see how we can be so confident in a calculation that doesn't use so much data
(Original post by ukdragon37)
That depends on the nature of your argument. For example Axiom of Choice => Zorn's Lemma => WellOrdering Theorem => Axiom of Choice is a wellknown circular set of results that stand independently from traditional foundations of set theory, and is unprovable from them unless you assume one of the equivalent (or stronger) forms. But that just reduces to the matter of what you pick as your axioms.
Yes they can.
Again, depends on the argument. A vacuous or tautological argument can have false assumptions.
Essentially the whole thing is ignoring the key foundation of deduction that justification is only relative to some set of axioms  within that axiomatic system the deductions could be perfectly sound, albeit perhaps open to dispute of matters of opinion (choice of axioms). In a carefully constructed logical system even infinite proofs can be shown to sound, and there are plenty of philosophical justifications for why one may like to choose to adopt that instead of the more classical systems we are used to that does not allow infinite proof.
Tl;dr: What you said appears to be a trilemma only because you are confining yourself to classic logic, and even then it's a false one since it ignores the fact that arguments are justified only relative to some axioms.
All cheeses are purple.
All iPods are cheese.
Therefore all iPods are purple.
This is a valid argument and is "true" relative to the assumptions that all cheeses are purple and all ipods are cheese... But wouldn't it be absurd to claim that the conclusion was ultimately justified (that is, it is appropriate to believe it)? If any argument we make relies upon something that we don't check to be true  we just assume it  then how can we claim to be justified when saying that any proposition is true or false? Our argument for its truth or falsity is necessarily based upon premises that we haven't verified.
If we have a reason to believe that one axiom is "better" or "truer" than another, then aren't we really justifying it? If so, then it isn't really an axiom, is it?Last edited by Implication; 01062013 at 18:30. 
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Problem 200 * / **
Three positive integers are chosen from the first 2k positive integers. What is the probability that the numbers chosen can form the sides of a triangle? What is the limiting probability as k tends to infinity? 
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 01062013 19:24
(Original post by jack.hadamard)
The second part can be done in three lines by using only value of Hurwitz's zeta function.
Solution 193
Rewrite .
Suppose . Then, , and . This implies that there exists prime such that , which is a contradiction.
Suppose . Clearly, , which is a contradiction again.
Now, let . Hence, ; thus there exists prime such that  contradiction.
Therefore, there are no nontrivial solutions to this diophantine equation. 
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Oh btw if anyone is thinking about life afar STEP, summer maths or something to do right now, then I've just found this really great website talking through all of the awesome integration techniques!
It's quite rare to get something that explains these techniques in a functional rare rather than via rigorous logic (which isn't particularly necessary for practical purposes) so enjoy! 
ukdragon37
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 01062013 23:18
(Original post by Implication)
I think I got a little confused in my last post, but I'll try to explain my thoughts again now. It may well be that I just don't understand, though.
Say we've measured the stress levels of managers and floor staff in a sample of shops and found that, in general, managers are more stressed than floor staff. We can do our stats to compute the probability that the difference in stress measurements between managers and floor staff is merely due to random chance. But, without any inductive inferences, we can't make any judgement about what did cause this difference: we can only decide whether it was likely to be due to random chance or a real variable difference. If it is likely to be due to a real variable, the stats still hasn't told anything about that variable beyond the fact that it isn't random chance.
H_{0}: There is no evidence to suggest being a manager affects stress levels any more than random chance.
H_{1}: Being a manger does significantly affect stress levels.
And you can then carry out a statistical test which determines which hypothesis is correct, as it was set out to do, but that as it should does not guarantee at all why you would find out anything beyond the hypotheses that are stated. If you then conjecture that stress levels are to do with shouting at subordinates, you could then frame that as appropriate hypotheses and you test again. What the statistical test guarantees you is that once you have found the a variable which you can show it is independent of random chance at , say, the 95% confidence level, then you know that there is a 95% chance that particular variable does indeed cause the phenomenon. If that wasn't the case then you just have to keep trying with another variable.
(Note that if you find H_{0} is the acceptable hypothesis then that does not mean being a manager does not affect stress levels, only there wasn't evidence of such from the data.)
XKCD demonstrates it quite nicely:
Spoiler:Show
What's the flaw here?
Essentially, yes you are right that you have to use some inductive experience to guess the correct variable to test, but there is nothing wrong with that  once you performed the test that gives you a purely deductive guarantee it is correct up to some confidence level.
EDIT: To those that are more pedantic than me, yes I know I am being sloppy with regards to the null hypothesis and interpretation of probability.
(Original post by Implication)
With regard to the icebergs, wouldn't our confidence interval necessarily rely on assumptions and inductive inferences? We may well have measured 1 000 000 icebergs and done our stats to find that we can be 99.7% certain that all icebergs are cold, for example... But what if there is a different type of underground iceberg that forms under pressure  and at higher temperatures  that we didn't know about then and is subsequently discovered? The stats may well have told us that the probability was 99.7%, but, without sounding like an excitable philosophy freshman with an obsession with Descartes and his method of doubt, were we really in a position to make that call when we had no way of knowing what information and data we were missing regarding icebergs in the first place? Our statistical calculation didn't (and couldn't have) taken this into account, so how can we ever have really been so sure in the first place?
In the former case, there is no statistical method which would have allowed you to obtain the figure anyway from measuring a subset of the population of icebergs, unless that population is finite and we know it (which would take into account the ones underground) or estimate it (which would exclude the possibility of the ones underground). The accuracy of the 99.7% figure is dependent on the accuracy of that estimate. If the population is infinite then naturally it is impossible to deduce (confidently or otherwise using statistics) that all icebergs are cold, unless there is some additional deductive evidence (such that the underground of Earth is proven to be inhospitable to icebergs).
If instead you meant that we are 99.7% sure that the next one we discover will be cold (i.e. a probability) then you should be aware that probability based on prior trials is only a relative probability based on exactly that, preexisting results. It could well go down if new information comes to light (a hot iceberg gets discovered, a new way of discovering icebergs is discovered, etc. etc.), and further it presumes that what we have observed is a fair sample of the population (here it may be not as underground icebergs are more difficult to observe). This is the case that usually gets twisted in the media as "we are X% sure every Y is Z" because they ignore the crucial assumptions.
In other words you are accusing statistics of something it's not guilty of.
(Original post by Implication)
I think that is exactly the point of the problem, aside from a difference in usage of the term "justification". Any argument we make necessarily relies upon some assumptions; axioms, doesn't it? So whether or not an argument is valid is entirely relative to these axioms.
All cheeses are purple.
All iPods are cheese.
Therefore all iPods are purple.
This is a valid argument and is "true" relative to the assumptions that all cheeses are purple and all ipods are cheese... But wouldn't it be absurd to claim that the conclusion was ultimately justified (that is, it is appropriate to believe it)? If any argument we make relies upon something that we don't check to be true  we just assume it  then how can we claim to be justified when saying that any proposition is true or false? Our argument for its truth or falsity is necessarily based upon premises that we haven't verified.
However it sounds like again you are accusing deduction for something which it is not. Justifiable (i.e. absolutely true) results from deductive reasoning is always in the form of "If X then Y", with X possibly being some axioms, or just "True", and this is justified absolutely because the task of the reasoning is to actually demonstrate it is a tautology. Note that all deductive reasoning is claiming to do is assert "if you believe in this, perhaps inductively, then you get that". It does not say "you should believe in this", if "this" is not a tautology.
In fact your example is actually defective because you have not "discharged all the assumptions". You have actually deduced "If all cheeses are purple, then if all iPods are cheese, then all iPods are purple". This as a whole is justified perfectly well.
(Original post by Implication)
If we have a reason to believe that one axiom is "better" or "truer" than another, then aren't we really justifying it? If so, then it isn't really an axiom, is it?Last edited by ukdragon37; 02062013 at 00:04. 
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 02062013 11:03
(Original post by Lord of the Flies)
Spoiler:ShowWoah! Just realised that problem 142 can be tweaked to form an algorithm which yields the successive values of . Firstly, we can solve the Basel problem using the result:
Noting that we get
Now for the general step define for and where
the fact that can be written as follows from the formula in solution 142 + induction
in other words, the only thing we need to do to evaluate the next value of is integrate a polynomial to get
For instance, integrating gives hence
Incidentally this also shows that for any for some .
Let ,
(1)
First we observe that (2) by letting .
By using (2) as well as making trigonometric substitutions,
(3)
Now, notice that in the proceeding working, we do not use any sophisticated techniques but only notation for special functions. Note that where is the digamma function.
Let ,
Let ,
(4)
Now we note that (5)
Equating (4) and (5) and letting
By differentiating we get that (6)
Using (6) and equating with (3),
Using (1),
Note also that, from this result, combined with the fact that , we get the delightful result that .Last edited by Jkn; 02062013 at 12:54. 
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 02062013 16:33
(Original post by Jkn)
Problem 195*/**
Prove that there exists infinitely many arithmetic progressions of 3 distinct perfect squares.
We must have:
Now, let:
Now, we can clearly choose an infinite number of even integers a and b such that this equation is satisfied, and if both a and b are even integers then n will also be an even integer. This satisfies all the conditions and so there are infinitely many arithmetic progressions of three perfect squares.
Problem 201: *
A stick of unit length is bent at a point along its length, with each such point being equally likely. The stick is then placed on a horizontal surface so that it forms a rightangled triangle, with the shorter edge of the stick being perpendicular to the ground. The angle of elevation of the longer edge of the stick with the ground is . Prove that the expected value of is given by:
Last edited by DJMayes; 02062013 at 18:15. 
bananarama2
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 02062013 17:16
(Original post by DJMayes)
Problem 201:
A stick of unit length is bent at a point along its length, with each such point being equally likely. The stick is then placed on a horizontal surface so that it forms a rightangled triangle, with the shorter edge of the stick being perpendicular to the ground. The angle of elevation of the longer edge of the stick with the ground is . Prove that the expected value of is given by:

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 02062013 17:29
(Original post by DJMayes)
Problem 201:
A stick of unit length is bent at a point along its length, with each such point being equally likely. The stick is then placed on a horizontal surface so that it forms a rightangled triangle, with the shorter edge of the stick being perpendicular to the ground. The angle of elevation of the longer edge of the stick with the ground is . Prove that the expected value of is given by:
First question  Is this a */**/*** problem?
Second question  Is this stick being dropped from the Eiffel Tower? 
bananarama2
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 02062013 17:30
(Original post by Zakee)
First question  Is this a */**/*** problem?
Second question  Is this stick being dropped from the Eiffel Tower?
This is going to become a TSR Meme 
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 02062013 17:51
(Original post by bananarama2)
I didn't expect this to be so pure
(I like trying to make the few questions I post here feel fairly similar to the sort of thing you'd get on STEP. Have I managed that with this one?)
(Original post by Zakee)
First question  Is this a */**/*** problem?
Second question  Is this stick being dropped from the Eiffel Tower? 
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 02062013 18:02
Solution 201
Let x be the length of the short piece.
We have
IBP
I left out the sub and rearrangement. I completely understand why LOTF does it now.
And then you multiply by two.
Footnote: The only stats I know is S1 and that from reading about QM, so I just basically made it up from what I knowLast edited by bananarama2; 02062013 at 18:19.
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