Motomuku

Edit summaries

By avoiding to use edit summaries, you make it very difficult to understand the reason behind your edits. In some cases, such as here and here I could not see any reason for your removal of content and citations, and had to revert to a previous version.

We appreciate your contributions to Wikipedia, but please be sure to use an edit summary on all your edits to avoid misunderstandings. Thank you. Owen× ☎ 13:36, 3 July 2009 (UTC)

I found a counterexample to your conjecture!

See Deficient number —Preceding unsigned comment added by 88.110.199.11 (talk) 23:49, 9 October 2009 (UTC)

I appreciate your letting me know. It is an intersting fact. Motomuku (talk) 17:29, 21 October 2009 (UTC)

Conjectures

I saw your conjectures at Abundant number and Deficient number:

Every integer of the form 11340m+ 8505 up to 10¹⁴ is abundant. It is not known whether every integer of this form is abundant.

Every integer of the form 12m+1 up to 10⁷ is deficient. It is not known whether every integer of this form is deficient.

There are infinitely many abundant numbers of the form 12n + 1; 367694231905 is an example. In fact such numbers have positive density!

Every number of the form 11340n + 8505 is divisible by 2835 and hence is abundant, since sigma(2835) = 5808 > 2 * 2835.

CRGreathouse (t | c) 01:18, 11 November 2009 (UTC)

Smallest example: 5391411025 = 5² × 7 × 11 × 13 × 17 × 19 × 23 × 29. More generally, there are infinitely many abundant integers of the form an + b for any a ≥ 1. CRGreathouse (t | c) 21:14, 3 December 2009 (UTC)

Iannucci's results re: 29 and 89

I removed your comment on Perfect number that neither 29 nor 89 is the special prime q. Iannucci proved this (1999; Lemma 12) under the assumption that N has at most one prime divisor greater than 10^4. This was a major step toward his proof that no such odd perfect number exists. But this is not known to hold generally for OPNs.

If you had something else in mind, please explain.

CRGreathouse (t | c) 19:50, 13 November 2009 (UTC)

Results by Panaitopol

${\frac {x}{\ln x-28/29}}<\pi (x)<{\frac {x}{\ln x-1.11}}$

for x ≥ 3299.

Interesting. I may have been wrong in removing this reference from the article; it's certainly an interesting result. (I'm not sure whether to replace the older reference, keep both, or keep only the older one.) But wait, it doesn't seem to work:

$2686=\pi (24121)>{\frac {24121}{\log 24121-1.11}}=2685.829\ldots$

Other points of failure: 24137, 24151, 24181, 24203, and 24251.

I see in the abstract that the author claims something like (depending on his definition of A)

$\pi (x)<{\frac {x}{\log x-1.08366}}$ for $x>10^{6}$

but this also seems to fail, starting at 1179427.

CRGreathouse (t | c) 16:12, 25 November 2009 (UTC)

Yes, you are right. Do you know what the best result is in this direction?

Motomuku (talk) 14:10, 1 December 2009 (UTC)

Not offhand. I'll let you know if I find anything.

CRGreathouse (t | c) 17:55, 1 December 2009 (UTC)

I haven't got the paper, but do you think this statement is provable?

$\pi (x)<{\frac {x}{\ln x-4}}$

for x ≥ 55.

As your reference shows, this was proved almost 70 years ago. In fact for any k > 1, there is an N such that for all n > N,

$\pi (x)<{\frac {x}{\ln x-k}}$

For k = 1.11, N is greater than the 3299 your source apparently claims, but there is some N that makes it true—the least is probably close to 24251. (If not, it's at least a billion.)

CRGreathouse (t | c) 20:05, 1 December 2009 (UTC)

Who proved it ?

For any k > 1, there is an N such that for all n > N,

$\pi (x)<{\frac {x}{\ln x-k}}$

Motomuku (talk) 07:14, 2 December 2009 (UTC)

I don't know, offhand; it's an old result. I'll look it up in H&W tonight to see if it's mentioned. More is true, actually: for every ε > 0, there is an N such that for all n > N,

${\frac {x}{\ln x-(1-\varepsilon )}}<\pi (x)<{\frac {x}{\ln x-(1+\varepsilon )}}$

CRGreathouse (t | c) 17:37, 2 December 2009 (UTC)

That is an another intersting result. Could you leave an evidence why that is true? Motomuku (talk) 06:56, 3 December 2009 (UTC)

It of course subsumes the other result. It follows, I suppose, from de la Vallée-Poussin's proof that Legendre's constant is equal to 1. CRGreathouse (t | c) 15:49, 3 December 2009 (UTC)

Possibly also worth mentioning: if I haven't made any mistakes in my derivation, the 'constant' k is asymptotically 1 + 1/log n + O((log log n/log n)²). Amusingly, using this backward, it suggests that Legendre was considering numbers around 155,304 when he conjectured the value of 1.08366. CRGreathouse (t | c) 18:50, 3 December 2009 (UTC)

Yes, this does follow from Legendre's constant. I didn't know about this constant. Thanks. For every ε > 0, there is an N such that for all n > N,

${\frac {x}{\ln x-(1-\varepsilon )}}<\pi (x)<{\frac {x}{\ln x-(1+\varepsilon )}}$

Hilbert's 10th

I removed your line

Susumu Hayashi, professor at Kyoto University, researched on Hilbert and concluded that Hilbert was asking for the consistency of real numbers. This was the goal of Gaisi Takeuti but still is an open question.

I don't know if Susumu Hayashi thought this, or if Gaisi Takeuti tried to prove it -- you haven't given any references. But this has not been an open problem for 60 years! Tarski solved the first-order case in the 1950s, and Tait solved the second-order version (stronger than Hilbert asked for) in the 1960s.

But none of this really matters, because Hilbert *wasn't* asking about real numbers. He was asking about rational numbers, which is clearly equivalent to asking about natural numbers or integers.

CRGreathouse (t | c) 23:17, 21 December 2009 (UTC)

Speedy deletion nomination of Wataru Takada

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Hilbert's 2nd

Listen, I think we got off on the wrong foot on this issue. First, I was for some reason confusing the 2nd and the 10th -- see section title above. Second, I'm a little sensitive to unexplained major changes like this, and tend to revert on sight. I've seen way too many edits changing a small detail that ruins the whole point of the sentence/paragraph/etc. I suppose I didn't WP:AGF properly; my apologies.

I brought it up on WT:MATH and it seems plausible. None of the other editors had seen anything saying that Hilbert meant real numbers, but if you have a reference it would probably be good to add it to the article. At the least, bring it up on the article Talk page and the editors there will find a way to integrate it into the page.

Because there are many points here, this should probably not be discussed in the lede but in another section, possibly its own section. This would certainly require one or more sources! Possible points: which did Hilbert mean, Tarski's results on decidability of first-order reals and geometry, Tait's result on second-order reals, Goedel's result on the undecidability of integers, Richardson's result on the undecidability of reals, ...

CRGreathouse (t | c) 21:09, 22 December 2009 (UTC)

Ishikawa's theorem

I removed the following fromPrime-counting function:

Heihachiro Ishikawa proved the following in 1934.

For integers x≥y≥2 and x≥6,

$\pi (xy)-\pi (x)>\pi (y)$ holds.

This result is interesting, but not appropriate for the article. It's also not very significant if it was discovered as late as 1934. It's a strengthening of Bertrand's postulate, but not as strong as Ramanujan's 1919 version of Bertrand's postulate.

Of course the first ≥ can safely be replaced with a , in the theorem.

CRGreathouse (t | c) 05:02, 8 January 2010 (UTC)

Previous appearance as a Wikipedia editor

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