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May 20

Magic square for base 2 Fermat pseudoprimes

There is a 3*3 magic square of primes:

17	89	71
113	59	5
47	29	101

Is there a 3*3 magic square of base 2 Fermat pseudoprimes, if no, is there a magic square (with any order) of base 2 Fermat pseudoprimes? 118.170.50.186 (talk) 12:17, 20 May 2024 (UTC)

May 22

Finding a list of "nice" angles

Hi. Angles such as 30, 45, 60 degrees are commonly used. I am currently using a very inefficient and convoluted process to find similar angles:

1. Draw a right triangle with height 4.0000 and the top angle as 39.0000 degrees. The bottom is then 3.23914.

2. Open up an excel sheet and populate the first column with values from 38.00 to 40.00 in 0.01 degree increments.

3. In the second column input the equation =tan(RADIANS(A1))*4.0000

4. Manually look through the second column to find a "nice" value (using human heuristics), which is 3.25999604

5. Calculate ROUND(3.25999604, 3)/4, which is 0.815

6. Now go back to the original right triangle, and change the angle to =ATN(0.815)

7. The right triangle now has height 4.0000 and the top angle as 39.1800 degrees, and bottom 3.2600

Is there a way to calculate all such "nice" angles in advance in the form of a table? If that were possible, there would be a row with the values: "tan(RADIANS(39.18)) = 0.815". So in step one, I would input height 4.0000, look in the table for an angle close to 39.0000, which is 39.1800, then input =ATN(0.815) as the angle. In that case, I would not need to open Excel and perform steps 2 to 6.

I am not a programmer but I imagine that it should be possible to write a short script to generate all such "nice" angles from 0 to 360 degrees.

Thank you and have a nice day. OptoFidelty (talk) 21:02, 22 May 2024 (UTC)

Perhaps Exact trigonometric values might be of use to you? NadVolum (talk) 22:19, 22 May 2024 (UTC)

One way to approach it would be to graph the function. So e.g. draw a graph of y = 4 tan(x) in a graph package. Excel e.g. though I've not used it for years so don't know how good its graph drawing is now.

Your "nice" angles then can be found when the graph crosses through, or close to, points on your "graph paper", marked to the degree of precision you desire. You might then be able to find these points programatically, or visually, or a combination of both from the graph. --2A04:4A43:90AF:FC35:6CCB:C14C:3818:FDA2 (talk) 23:30, 22 May 2024 (UTC)

Thanks, guys. Exact trigonometric values is very close to what I am looking for. OptoFidelty (talk) 15:33, 23 May 2024 (UTC)

Does the algebraic (roots) notation exist for the trigonometric values of all angles with integer degrees? 61.224.168.169 (talk) 11:44, 24 May 2024 (UTC)

By the Gauss–Wantzel theorem, the angles of integral degree with algebraic trigonometric values are precisely those that are a multiple of 3°.  --Lambiam 04:39, 25 May 2024 (UTC)

It is not quite clear to me from this one example what makes an angle "nice". Suppose I draw a right triangle with height 7.5 and top angle 52°. I get 9.59956... for the width. This is close to 9.6. Is it close enough for 52° to be nice?

Is the following nice: height = 7.9735, angle = 9.0941°, width = 1.2763? If not, why not?  --Lambiam 21:02, 23 May 2024 (UTC)

@Lambiam The triangle I ended drawing has height 4, top angle 39.1800 degrees, and bottom 3.2599960396627881356849171185262635206102761449606762736889760643... long.

On a schematic with a 2 decimal point formatting, the rounded numbers would be height 4.00, top angle 39.18 degrees, and bottom 3.26. In that case, the height and angle are exact, and the bottom number is 0.000121482% off from the exact value.

With height = 7.9735, angle = 9.0941°, width = 1.2763, the 2 digit rounded values would be 7.97, 9.09, 1.28. And the rounding error would be slightly larger than my example.

"niceness" is entirely subjective and varies from person to person, and from context to context. In this case, it's basically a personal shorthand word I use to describe "when you round the number to X number of decimal points, the rounded number is less than Y% off from the exact measurement".

The value of X is determined by the exact drafting standard that I am told to draft in. It commonly varies from 1 to 3. the value of Y is, again, subjective. I personally like to keep it "small", but there is no objective measure on how "small" Y needs to be. OptoFidelty (talk) 21:23, 24 May 2024 (UTC)

The values of X and Y need to be fixed if you want to construct a table. The range of heights considered also needs to be made finite. Then it is a somewhat trivial exercise to code an algorithm that enumerates all possibilities and outputs the nice ones.  --Lambiam 04:51, 25 May 2024 (UTC)

Thank you. Does the value of Y need to be fixed or can it be estimated from the desired number of table rows?

For example, if X = 2 and the desired table size is 1000, can Y be estimated from that?

Or maybe when given the desired table size, let's say 1000, it's easier just to loop through all possible angles values (360 * 10^X), then just keep the "best" 1000 values then it's done. Y isn't actually needed in that case. OptoFidelty (talk) 13:11, 25 May 2024 (UTC)

The notion of "desired table size" is a new element. You can do either – keep all that are nice, or keep the best N, whether nice or not. You can also keep the best N but discard any that are not nice. Which works best for you depends on what you use these tables for.  --Lambiam 18:05, 25 May 2024 (UTC)

May 23

Differential Equation

Does the differential equation x² * d²x/dt² = k have a name? (All I've figured out about this so far is that I don't remember enough about differential equations. I'm not getting anything on solving it except errors.) Thank you. RJFJR (talk) 02:45, 23 May 2024 (UTC)

As to a solution, you could guess that one might be some power of $t$ and accordingly substitute $x=At^{p}$ (where $A$ and $p$ are constants), then solve for $p$ and then solve for $A$. catslash (talk) 09:02, 23 May 2024 (UTC)

Then, since nothing in the equation depends on the absolute value of $t$, you could apply an arbitrary time-shift to get a slightly more general solution $x=A(t-t_{0})^{p}$. catslash (talk) 09:20, 23 May 2024 (UTC)

You're right. Thank you. p=2/3, I was not expecting that. I appreciate it. RJFJR (talk) 14:14, 23 May 2024 (UTC)

This is an autonomous second order equation. If you write ${\ddot {x}}-{\frac {k}{x^{2}}}=0$ and multiply with ${\dot {x}}$, you find ${\frac {d}{dt}}({\frac {1}{2}}{\dot {x}}^{2}+{\frac {k}{x}})=0$ so ${\frac {1}{2}}{\dot {x}}^{2}+{\frac {k}{x}}=C$ for some constant $C$. Solve for ${\dot {x}}$ and you get a separable first order equation. —Kusma (talk) 14:35, 23 May 2024 (UTC)

Thank you. I need to dig out the old textbook and start reading. RJFJR (talk) 19:25, 23 May 2024 (UTC)

catslash's solution is a one-parameter family (indexed by $t_{0}$) of very nice solutions, but in general you should be able to solve the initial value problem for any initial values of $x$ and ${\dot {x}}$, so you'll get a two parameter family. It is easy to show that the solution exists; you can get an implicit formula from Mathematica or other symbolic computation software. —Kusma (talk) 12:17, 24 May 2024 (UTC)

Autonomous system (mathematics)#Special case: x″ = f(x) gives $t$ as a two-parameter function of $x$, but this function looks uninvertable except for the choice of the parameter $C_{1}$ which makes it correspond to my guessed solution. catslash (talk) 22:43, 25 May 2024 (UTC)

May 27

Szekeres Conjecture

Is there something like Szekeres Conjecture, which is different from Erdős–Szekeres theorem? ExclusiveEditor _{Notify Me!} 19:00, 27 May 2024 (UTC)

The sum will never reach 2

I saw a reference to Zeno's paradoxes#Dichotomy paradox in a comic strip. The article does not mention the sum of 1, one half, one quarter, and so on. Where is that sum?— Vchimpanzee • talk • contributions • 22:34, 27 May 2024 (UTC)

The sum is 1 if you sum an infinite number of terms. Bubba73 ^{You talkin' to me?} 23:42, 27 May 2024 (UTC)

Would you believe 2? --142.112.143.8 (talk) 03:59, 28 May 2024 (UTC)

I was hoping to link the sum I was asking about from the Zeno's paradox article.— Vchimpanzee • talk • contributions • 14:37, 28 May 2024 (UTC)

Well, Google actually gave me 1/2 + 1/4 + 1/8 + 1/16 + ⋯.— Vchimpanzee • talk • contributions • 14:37, 28 May 2024 (UTC)

Our article does not mention a sum, but an infinite regression of tasks. Each task has a subtask that must be completed before the whole task can be completed. This is (in Zeno's analysis) as impossible as the task of enumerating all unit fractions in order of magnitude, so that 1/1000 has to come before 1/999 – you can't even start.  --Lambiam 15:44, 28 May 2024 (UTC)

Do you think the Zeno article should mention the other?— Vchimpanzee • talk • contributions • 18:58, 28 May 2024 (UTC)

To be more precise, the section Dichotomy paradox does not mention this sum. Elsewhere, in the introductory paragraph of the section Paradoxes, it is stated that Zeno's paradoxes are often presented as an issue with the sum of an infinite series, although none of the original ancient sources has Zeno discussing the sum of an infinite series. In my opinion it can be given a place in the analysis of the Achilles and the Tortoise paradox, but not so for Zeno's dichotomy paradox.  --Lambiam 20:08, 28 May 2024 (UTC)

May 29

What is 'lakh' and 'crore'? How can I understand Indian articles with strings such as 1,00,00, 1,50,00 and so on ?

sirs i really don't understand this lakh and crore business which has lately become very common on the internet apparently it is some kind of indian custom, indian reckoning please tell me how can you place the comma after two digits only (counting from the front) firstly how can you go beyond lakh and crore how can i reckon a number with eight digits 1,00,00,00 ??? one hundred lakh and how many crore ? if a crore is one tenth of a lakh, hold on, one lakh is just a hundred thousand ? so how much is a crode please?? if this 1,00,00 is a lakh entire sirs please forgive my discursiveness i was trying to read up each and everything on this subject and i could not put my head around it kindly direct me to any pertinent source where i can understand lakh, crore, indian customary numbers 2601:481:80:6E60:6C4B:56AE:F2AB:2844 (talk) 23:55, 29 May 2024 (UTC)

We have articles on Lakh and Crore. Do they answer your question? They seem to be a feature of the Indian counting system, and not common in other countries. --RDBury (talk) 00:27, 30 May 2024 (UTC)

I think I have studied very well and I am understanding it now.surely this will improve my scores in JEE.the invigilators and examiners will be very pleased with my fast reckoning. thank you sirs 2601:481:80:6E60:6846:DCBF:5125:F19D (talk) 05:18, 31 May 2024 (UTC)

If you see commas in unexpected places, the easiest is to ignore them. 12,34,567 is the same number as 1234567.  --Lambiam 06:20, 30 May 2024 (UTC)

Kindly don't be writing just any x y z, how can it be same amount if the crode and lakh is arranged in a different manner, since a crode and lakh have values each of their own, they cannot be mingled around or preponed 2601:481:80:6E60:6846:DCBF:5125:F19D (talk) 05:17, 31 May 2024 (UTC)

They are interconvertible values. One crore is equal to 100 lakh, and also equal to 10 million. So 3.5 crore + 7 lakh = 35000000 + 700000 = 35700000, which you can also write as 3,57,00,000 or as 35,700,000 or as 35 700 000.  --Lambiam 09:27, 31 May 2024 (UTC)

May 30

A proof attempt for the transcendence of ℼ

The proposition "if $a$ is rational then $a$ is algebraic" is comprehensively true,
and is equivalent to "if $a$ is inalgebraic then $a$ is irrational" (contrapositive).

My question is this:
The proofs for the transcendence of $\pi$ are of course by contradiction.
Now, do you think it is possible to prove somehow the proposition "if $\pi$ is algebraic then $\pi$ is rational", reaching a contradiction?
Meaning, by assuming $\pi$ is algebraic and using some of its properties, can we conclude that it must be algebraic of degree 1 (rational) – contradicting its irrationality?

I know the proposition "if $a$ is algebraic then $a$ is rational" is not comprehensively true (${\sqrt {2}}$ is a counterexample),
but I am basically asking if there exist special cases $a\in \mathbb {R}$ such that it does hold for them. יהודה שמחה ולדמן (talk) 18:48, 30 May 2024 (UTC)

There are real-valued expressions $E$ such that the statement "if $E$ is algebraic, $E$ is rational" is provable, but this does not by itself establish transcendence. For example, substitute ${\tfrac {22}{7}}$ for $E.$ Given the irrationality of $\pi ,$ proving the implication for $\pi$ would give yet another proof of the transcendence of $\pi$. I see no plausible approach to proving this implication without proving transcendence on the way, but I also see no a priori reason why such a proof could not exist.  --Lambiam 19:24, 30 May 2024 (UTC)

May 31

Meridional Radius of Curvature

Hi y'all.

$M(\varphi )d\varphi ={\frac {(ab)^{2}}{{\big (}(a\cos \varphi )^{2}+(b\sin \varphi )^{2}{\big )}^{\frac {3}{2}}}}d\varphi ={\sqrt {(a\sin \beta )^{2}+(b\cos \beta )^{2}}}d\beta =??(\beta )d\beta .$

(φ, β = geodetic, reduced latitudes)

If ${\frac {(ab)^{2}}{{\big (}(a\cos \varphi )^{2}+(b\sin \varphi )^{2}{\big )}^{\frac {3}{2}}}}=M(\varphi )$ equals the "meridional radius of curvature", then what does

${\sqrt {(a\sin \beta )^{2}+(b\cos \beta )^{2}}}\,$ equal ("reduced meridional radius of curvature"?) and what is its symbol (_rM(β)? )?

--2601:19C:4A01:7057:4C27:AD22:B7E2:D04A (talk) 15:35, 31 May 2024 (UTC)