Wikipedia:Reference desk/Archives/Mathematics/2021 June 5

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June 5

An elementary proof is there (2)

For the eqn. [math] ab + c = t [/math] where [math]a,b,c[/math] are three consecutive numbers. show that [math] t [/math] will never be a perfect power? — Preceding unsigned comment added by Rajesh Bhowmick (talk • contribs) 03:28, 5 June 2021 (UTC)

Is this a homework problem?

I assume you mean consecutive integers, not numbers.

And your premise is trivially false for the consecutive integers -1, 0, 1. ~Anachronist (talk) 04:26, 5 June 2021 (UTC)

Wikipedia is an encyclopedia. Please read What Wikipedia is not. We are specifically not a forum for discussion or a place to publish original research.  --Lambiam 08:34, 5 June 2021 (UTC)

From the relevant section of that page (emphasis removed and added):

4. Discussion forums. Please try to stay on the task of creating an encyclopedia. You can chat with people about Wikipedia-related topics on their user talk pages, and should resolve problems with articles on the relevant talk pages, but please do not take discussion into articles. In addition, bear in mind that article talk pages exist solely to discuss how to improve articles; they are not for general discussion about the subject of the article, nor are they a help desk for obtaining instructions or technical assistance. Material unsuitable for talk pages may be subject to removal per the talk page guidelines. If you wish to ask a specific question on a topic, Wikipedia has a Reference desk; questions should be asked there rather than on talk pages.

I believe that means that this page falls within an exception to the prohibition on general discussion.

And Wikipedia:No original research says that it applies only to Wikipedia articles. --116.86.4.41 (talk) 10:55, 5 June 2021 (UTC)

Note that this is just a restatement of the previous question. Perhaps it was a homework problem to show the two are equivalent. --RDBury (talk) 11:44, 5 June 2021 (UTC)

It comes from this problem being posed at Quora, where this OP posted a solution for the rather special case that c is a multiple of 4. They tried multiple times (in vain, each time reverted) to add a link to it in our articles Catalan's conjecture and Fermat's Last Theorem before coming here.  --Lambiam 12:29, 5 June 2021 (UTC)

Bicycle tracks

While I was pushing my bike across a sandy beach, it occurred to me that if I pushed it around in a circle, the front and back wheels would describe concentric circles. Is there any other case, other than the trivial one of the straight line, where the front and back wheel tracks would be the "same type of curve", by which I suppose I mean the same curve scaled, or shifted, or rotated, or any other sensible interpretation of "same type" that would yield an interesting solution. Pushing the bike in a sine-wavy path produces two sine-wavy tracks, but if the front track is exactly a sine wave then it seems to me that the back one isn't exactly. 2A00:23C8:7B08:6A00:1D58:69CA:8C26:5FEC (talk) 21:20, 5 June 2021 (UTC)

A simple mechanical model is that of the Prytz planimeter. If the leading point is drawn along a straight line, the point being dragged follows a tractrix. I can confirm (by numerical integration of the differential equation) that the wavy follow-track of a sinusoidal lead-track does not exactly have the sine shape. In particular, the extrema occur earlier than the midpoints between two zero crossings.  --Lambiam 11:43, 6 June 2021 (UTC)

Mathcurve.com has an article on the generalized tractrix, which they call a tractory. They don't cover the case where the tractory and its generating curve are similar though. --RDBury (talk) 16:11, 6 June 2021 (UTC)

The first problem in this charming book concerns the question of finding a path for a bicycle so that the rear wheel only travels where the front wheel has been. (A straight line is one such path, but there are others.) It explores a number of variations, and includes the following papers as references: [1] [2] [3] [4] [5] [6]. --JBL (talk) 17:45, 7 June 2021 (UTC)

Well. That surprises me that the front and rear tracks could be identical, other than the straight-line case. I find that quite hard to visualise. 2A00:23C8:7B08:6A00:19FB:9DD4:4536:E95A (talk) 22:00, 9 June 2021 (UTC)

They are quite wild. The third link I shared (the article in CMJ by D. Finn) has on its first page (visible even without a subscription) a portion of such a path. As the path extends, the wiggles become more and more extreme -- see Figures 3 and 16 of [7]. --JBL (talk) 22:17, 9 June 2021 (UTC)

In the path depicted on the first page of your third link, [8], "Can a bicycle create this track?", the answer to which I assume from your reply is "Yes", do you have any idea of the distance between the wheels, relative to the size of the features in the path? 2A00:23C8:7B08:6A00:3DBC:CE70:1CBC:5920 (talk) 21:57, 10 June 2021 (UTC)

At any time, the tangent to the curve at the location of the rear point intersects the curve at the location of the front point; the distance between these is constant. Take the first shallow up bump, at about 30% of the width of the figure, going left to right. From the top of the bump, the horizontal tangent intersects the curve many times, but the relevant point of intersection is the second one, where the curve is sloping down. You can do just the same for the next, much larger up bump. Another point is just at the start of that next larger up bump. It has a point of inflection, and the tangent there, going up at an angle of about 1 radian, is actually also tangent at the front point of intersection. In all cases, the distance is about 30% of the width of the figure.  --Lambiam 23:15, 10 June 2021 (UTC)

The full article can be downloaded from this page (click "Read the article"). The scale can be seen in Figures 4 and 5, corresponding almost exactly to the situations in the first and last of the cases I mentioned above.  --Lambiam 10:21, 11 June 2021 (UTC)