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Originally Posted by

**Deveno** your motivating observation is simple enough. but i point out [...]

I don't mind suggested simplifications. But the proof is still straightforward even if there are more elegant proofs. It doesn't involve any advanced notions in mathematics; all that is involved are checking some grammar school additions and then mathematical induction. As to the number of "base cases", again I don't claim elegance. Indeed, I was going to use the phrase "brute force" a couple of times in the proof, but I took it out because it's not needed to mention that I'm using "brute force" since it is obvious enough that I am.

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Originally Posted by

**Deveno** if i were a professor reading this proof (perhaps as some homework assignment), i would probably mark off

Mark off for what? Lack of elegance? Then you have to give elegance as a grading criterion. If I were grading proofs, I would grade solely on whether the proof is correct and supplies reasonable detail. I wouldn't mark for elegance unless elegance were STATED as a criterion.

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Originally Posted by

**Deveno** "unsupported assumptions". yes, we CAN find j,k such that 14,16,18,20 and 22 can be written as 3j+8k for some j,k, but that's part of what one is supposed to be proving.

Oh, come on, those have simple, finite verifications in grammar school addition. In a proof for grown up readers it's not needed to belabor such matters.

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you might as well say: "for every n > 13, we can obviously write n = 3j + 8k for non-negative j,k."

No, I can't just say that. It requires proving.

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it's not just a matter of elegance or simplicity. it's a matter of clarity. when i read your proof, it was not immediately obvious, what you were saying is true,

Then you weren't reading carefully at all. At every line I stated what assumption I was making, then I stepped through to conclusions from those assumptions. Then the assumptions are dischaged in the obvious way.

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in one line you take as an assumption "all 7 < q < r",

and then in the next line you talk about q in {3, 4, 5, 6, 7}, leaving one to wonder how you can assume q > 7 on one hand, and q = 3, on the other.

No, you misparaphrased.

I wrote:

"For the stong induction hypothesis, suppose for all q such that 7 < q < r, we have there are j and k such that 2q = 3j+8k.

Now for some q, let 2r = 2q + 8. So q < r. And, since r > 7, we have q > 2. Now if q is in {3, 4, 5, 6, 7}"

I didn't assume "all 7 < q < r". That doesn't make sense standalone in this context. What I ACTUALLY said is:

"suppose for all q such that 7 < q < r, we have there are j and k such that 2q = 3j+8k".

That is the strong induction HYPOTHESIS.

Also, we have from previous work that for SOME q we have 2r = 2q + 8.

Then, also, I narrowed down to what happens if that q is in {3, 4, 5, 6, 7}.

The method is perfectly clear. First I have a strong induction hypothesis that is a universal generalization on q (for all q, if 7 < q < r then we have that there are j and k such that 2q = 3j+8k). And I have an existential generalization q (there is a q such that 2r = 2q + 8.) Then I grabbed that q, took cases on it, and applied the earlier universal generalization on it too. Perfectly legal and clear.

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that kind of tangled variable reference ought to be avoided in a proof, it muddies the clarity of your argument.

No, it was clear, and economical use of variables. For all q, blah blah holds. And there is some q such that glab glab holds. Consider some q such that glab glab holds. Now infer about it with the fact that both blah blah and glab glab holds for it.

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you say "By weak induction on r, it's easy to show: If r > 3 then there is a q such that 2r = 2q+8you omit certain vital facts about q.

No, I didn't omit any vital facts about q. All I need there is exactly what I said: If r > 3 then there is a q such that 2r = 2q+8. Yes, I ommitted belaboring the weak induction that allows me to infer that if r > 3 then there is a q such that 2r = 2q+8, but that is because the induction there is quite routine.

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you omit certain vital facts about q. i can infer that what you meant was:

it's true for ANY integer r (or real number, even)

Oh, come on, from all the previous discussion and from the context of the proof, one can easily allow that I'm talking about natural numbers. But one doesn't even have to allow it, since I STATED I'm doing "induction on r", so OF COURSE r is a natural number. As grown ups, in such context, we don't pick on such nits as "you didn't say you're only working with natural numbers" when the context is clear enough that we are and when one says ANYWAY in such a context "INDUCTION ON".

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any other i've seen posted in this topic.

Would you please tell me specifically what post you consider to be an elegant, more or less self contained, and sufficiently detailed proof that 14 is the desired number?

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at one point you say "By weak induction on r, it's easy to show: If r > 3 then there is a q such that 2r = 2q+8."

think about that for a second. you are asking the reader to supply an inductive proof, which is then a lemma in YOUR proof.

How about you think for a second that mathematical proofs OFTEN mention things like "it's easy to see" or "the verification is routine" et. al when indeed it's quite easy to whip out a trivial bit of reasoning like the quite easy weak induction I'm referring to there.

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why, should i as the reader, be asked to prove part of your proof, when you are the one trying to prove something?

Because you can flash it through your head, or jot it quick on paper just about as fast it would take you to read it already typed out for you. Are you serious? You don't see that it's utterly common in mathematics that authors or givers of notes leave some too obvious steps with remarks such as "the weak induction here is routine"?

Look, do you have the slightest difficulty seeing how routine the weak inductions are? If not, then it's captious of you to make an issue of it.

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from a pedagogical point of view,

I don't claim to provide sparkling pedagogy. My point was merely to show that there is indeed a proof that 14 is the number and to use induction (and using both kinds) to do it.

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Originally Posted by

**Deveno** it's not sarcasm

That denial makes it worse, given that your tone definitely was sarcastic: "pretty straight-forward, eh?" along with you striving to be arch with "by the time one is finished, one has the feeling that SOMETHING has been proved," which sounds like some affectation you got from reading too many hack movie reviews: "By the time the credits roll, one has the feeling that there was some point or another in the making of this film..."