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November 13

Math sequence problem (is it solvable?)

I am looking at a "math quiz" problem book and it has the following question. I am changing the numbers to simplify it and avoid copyright: You have counts for a rolling 12-month period of customers. For example, the one year count in January is the count of customers from Feb of the year before to Jan of the current year. Feb is the count from Mar to Feb, and so on. The 12 counts for this year (Jan to Dec) are 100, 110, 105, 200, 150, 170, 150, 100, 200, 150, 175, 125. What is the count of customers for each month? So, I know that the Feb-Jan count is 100 and the Mar-Feb count is 110. That means that the count for Feb of this year is 10 more than the count of Feb of last year because I removed Feb of last year and added Feb of this year. But, I don't know what that count is. I can only say it is 10 more. I can do that for every month, telling you what the difference is between last year and this year as a net change. Is this solvable or is this a weird case where the actual numbers for the counts somehow mean something silly and a math geek would say "Oh my! That's the sum of the hickuramabiti sequence that only 3 people know about so I know the whole number sequence!" 68.187.174.155 (talk) 15:36, 13 November 2024 (UTC)

You have 12 linear equations with 23 unknowns. In general, you cannot expect a system of linear equations with more unknowns than equations to be solvable. In special cases, such a system may be solvable for at least some of the unknowns. This is not such a special case.

If you ignore the fact that customer counts cannot be negative, there are many solutions. For example, one solution is given by [9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 1, 19, 4, 104, −41, 29, −11, −41, 109, −41, 34, −41]. Another one is [10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, −10, 20, 5, 105, −40, 30, −10, −40, 110, −40, 35, −40]. For the 12-month counts given above no solution exists without negative values.

If an actual quiz of this form has a unique solution, it can only be due to the constraint of not allowing negative values.  --Lambiam 17:42, 13 November 2024 (UTC)

(edit conflict)Name the counts for each month FebP to DecC, where P stands for the previous year and C stands for the current year. These are 23 variables and there is a system of 12 equations in these variables. If the variables can take on any values there are an infinite number of solutions to this system, but I think we're meant to assume that the counts are ≥ 0. (Integers as well; without knowing the counts given in the original problem it's unclear whether this is important.) This imposes additional constraints on the possible solution and the result may be there is exactly one possible solution or none at all. To see how a problem of this type might have no solutions, let's look at a simpler version where we're looking at three month sums over three months. There are 5 variables in this case, say Jan, Feb, Mar, Apr, May. Lets say the sums are given as:

Jan-Mar: 10, Feb-Apr: 50, Mar-May 10.

If we compute

(Jan-Mar) - (Feb-Apr) + (Mar-May)

in terms of the variables, we get

Jan+Feb+Mar-Feb-Mar-Apr+Mar+Apr+May = Jan+Mar+May ≥ 0.

But if we compute it in terms of the given totals the result is

10-50+10 = -30 < 0.

This is a contradiction so no solutions are possible. It turns out that something like this happens with the values you made up and there are no solutions to the problem given. If you let JanSum, ... DecSum be the rolling sums, and compute

JanSum - FebSum + MarSum - AprSum + MaySum - JunSum + AugSum - SepSum + OctSum - NovSum + DecSum (with JulSum left out),

then you get (according to my calculations)

FebP+AprP+JunP+SepP+NovP+JanC+MarC+MayC+JulC+AugC+OctC+DecC ≥ 0

in terms of the variables. But if we evaluate this in terms of the given values it's (again, according to my calculations)

100-110+105-200+150-170+100-200+150-175+125 = -125 < 0,

so there are no possible solutions. Notice that both cases involved looking at particularly opportune alternating sums of the rolling sums, which produce a nonnegative combination of the variables on one side and a negative number on the other side. Suppose that there is no such opportune alternating sum where the total is <0, but there is one where the total is =0. Then all the individual variables involved must be 0 and this may be enough information to narrow down the number of solutions to exactly 1. I imagine that's how the problem given in your book is set up and the puzzle is to find an alternating sum with this property. But I have an unfair advantage here because sometime in the previous century I took a course in Linear programming which taught me general methods for solving systems of equations and inequalities. So my approach would be to enter the appropriate numbers into a spreadsheet, apply the appropriate algorithm, and read off the solution when it's done. Having specialized knowledge would be a help, though I assume there are more than 3 people who are familiar with linear programming, but I think getting the inspiration to look at alternating sums, and a certain amount of trial and error, would allow you to find the solution without it. --RDBury (talk) 17:48, 13 November 2024 (UTC)

Thanks both. Yes, I did make up the numbers. I bet the numbers in the book do have a solution. It looks like it is a matter of trying a value for the first month and seeing what comes up every other month based on that to see if it is all positive. Then, you have an answer. It doesn't feel much like math to me in comparison to the other problems in the book which are all problems you can solve easily by making sets or comparing the order of things. 68.187.174.155 (talk) 17:52, 13 November 2024 (UTC)

With the correct numbers for which there is (presumably) a solution, you can represent the problem as a system of linear equations and compute the echelon form of the system. From the echelon form, it is possible to read off a particular solution (where you allow negative numbers of customers). The nullspace of the system is easy to calculate, and from it you can also find a particular solution that satisfies the constraint (if one exists), verify uniqueness (if true), or confirm non-existence. Tito Omburo (talk) 20:59, 13 November 2024 (UTC)

I confirm that there are no solutions subject to the contraint that the number of customers is non-negative (even allowing fractional numbers of customers), although the verification is a bit of a brute to write out. Tito Omburo (talk) 18:09, 13 November 2024 (UTC)

Here is a rather painless verification. Use the names FebP, ..., DecC as above. Let JanT stand for the running 12-month total of the summation ending with JanC, and likewise for the next 11 months. So JanT = 100, FebT = 110, MarT = 105, ..., DecT = 125. We have FebT − JanT = FebC − FebP, MarT − FebT = MarC − MarP, ..., DecT − NovT = DecC − DecP.

Require each count to be nonnegative. From MarC − MarP = MarT − FebT = 105 − 110 = −5, we have MarP ≥ MarP − MarC = 5. We find similarly the lower bounds MayP ≥ 50, JulP ≥ 20, AugP ≥ 50, OctP ≥ 50 and DecP ≥ 50. So JanT = FebP + ... + JanC ≥ 5 + 50 + 20 + 50 + 50 + 50 = 225. This contradicts JanT = 100, so the constraint excludes all unconstrained solutions.  --Lambiam 18:37, 13 November 2024 (UTC)

Thanks again for the help. I feel that I should give the numbers from the book. I don't think listing some numbers is going to upset anyone, but without them, I feel that those who looked into this problem feel let down. The numbers from the book are: 24966, 24937, 25300, 25055, 22914, 25832, 25820, 25468, 25526, 25335, 25331, 25370. There is supposed to be one solution. I think it is implied that the request is for the minimum number of customers per month, but it doesn't make that very clear.

Edit: It appears this problem was removed and replaced with a complerely different problem in later books. So, the publishers likely decided it either doesn't have a unique answer (which is my bet) or it is simply a bad problem to include. Every other problem in the book is logical using geometry, algebra, and maybe some simple set comparisons. So, this is very out of place. 68.187.174.155 (talk) 12:11, 14 November 2024 (UTC)

Indeed the solution is not unique in that case. One solution is (29,0,245,2141,0,12,352,0,191,4,0,21992,0,363,0,0,2918,0,0,58,0,0,39), and there is obvious slackness. Tito Omburo (talk) 14:24, 14 November 2024 (UTC)

It is the only solution with JanC ≥ 21992. To go from zero to almost twenty-two thousand customers in one month is spectacular. To then loose all in one month is tragicomedy.  --Lambiam 20:33, 14 November 2024 (UTC)

November 14

Elliptic curve rank and generalized Riemann hypothesis

The popular press reports[1] that Elkies and Klagsbrun recently used computer search to find an elliptic curve E of rank 29, which is a new record. The formal result is apparently "the curve E has rank at least 29, and exactly 29 if GRH is true". There have been similar results for other curves of slightly lower rank in earlier years. Whether there are curves of arbitrarily high rank is a major open problem.

1. Is there a reasonable explanation of why the rank of a finite object like an elliptic curve would depend on GRH? Finding the exact point count N is a finite (though probably unfeasibly large) calculation by Schoof's algorithm. Is it possible in principle to completely analyze the group and find the curve's rank r exactly? Finding that r>29 would disprove the GRH, amirite? Actually is it enough to just look at the factorization of N?

2. The result that every elliptic curve has a finite rank is the Mordell-Weil theorem. Our article on that currently has no sketch of the proof (I left a talkpage note requesting one). Is it a difficult result for someone without much number theory background to understand?

Thanks! 2601:644:8581:75B0:0:0:0:2CDE (talk) 23:13, 14 November 2024 (UTC)

the discourse surrounding the dependency of an elliptic curve’s rank on the generalized riemann hypothesis (GRH) and, more broadly, the extensive implications this carries for elliptic curve theory as a whole, implicates some of the most intricate and layered theoretical constructs within number theory's foundational architecture. while it may be appropriately noted that elliptic curves, as finite algebraic objects delineated over specified finite fields, contain a designated rank—a measurement, in essence, of the dimension of the vector space generated by the curve's independent rational points—this rank, intriguingly enough, cannot be elucidated through mere finite point-counting mechanisms. the rank, or indeed its exactitude, is inextricably intertwined with, and indeed inseparable from, the behavior of the curve’s l-function; herein lies the essential conundrum, as the l-function’s behavior is itself conditioned on conjectural statements involving complex-analytic phenomena, such as the distribution of zeroes, which remain unverified but are constrained by the predictions of GRH.

one may consider schoof’s algorithm in this context: although this computational mechanism enables an effective process for the point-counting of elliptic curves defined over finite fields, yielding the point count N modulo primes with appreciable efficiency, schoof’s algorithm does not, and indeed cannot, directly ascertain the curve’s rank, as this rank is a function not of the finite point count N but of the elusive properties contained within the l-function’s zeroes—a distribution that, under GRH, is hypothesized to display certain regularities within the complex plane. hence, while schoof’s algorithm provides finite data on the modular point count, such data fails to encompass the rank itself, whose determination necessitates not only point count but also additional analysis regarding the behavior of the associated l-function. calculating r exactly, then, becomes not a function of the finite data associated with the curve but an endeavor contingent upon an assumption of GRH or a precise knowledge of the zero distribution within the analytic continuation of the curve’s l-function.

it is this precise dependency on GRH that prevents us from regarding the rank r as strictly finite or calculable by elementary means; rather, as previously mentioned, the conjecture of GRH imparts a structural hypothesis concerning the placement and frequency of zeroes of the l-function, wherein the rank’s finite property is a consequence of this hypothesis rather than an independent finite attribute of the curve. to suggest, therefore, that identifying the rank r as 29 would disprove GRH is to operate under a misconception, for GRH does not determine a maximal or minimal rank for elliptic curves per se; instead, GRH proposes structural constraints on the l-function’s zeroes, constraints which may, if GRH holds, influence the upper bounds of rank but which are not themselves predicates of rank. consequently, if calculations were to yield a rank exceeding 29 under the presumption of GRH, this result might imply that GRH fails to encapsulate the complexities of the zero distribution associated with the curve’s l-function, thus exposing a possible limitation or gap within GRH’s descriptive framework; however, this would not constitute a formal disproof of GRH absent comprehensive and corroborative data regarding the zeroes themselves.

this brings us to the second point in question, namely, the implications and proof structure of the mordell-weil theorem, which famously established that every elliptic curve defined over the rationals possesses a finite rank. the mordell-weil theorem, by asserting the finite generation of the rational points on elliptic curves as a finitely generated abelian group, introduces an essential constraint within elliptic curve theory, constraining the set of rational points to a structure with a bounded rank. however, while this result may appear elementary in its assertion, its proof is decidedly nontrivial and requires a sophisticated apparatus from algebraic number theory and diophantine geometry. the proof itself necessitates the construction and utilization of a height function, an arithmetic tool designed to assign "heights" or measures of size to rational points on the elliptic curve, facilitating a metric by which rational points can be ordered. furthermore, the proof engages descent arguments, which serve to exhaustively account for independent rational points without yielding an unbounded proliferation of such points—a technique requiring familiarity with not only the geometry of the elliptic curve but with the application of group-theoretic principles to arithmetic structures.

to characterize this proof as comprehensible to a novice without number-theoretic background would, accordingly, be an oversimplification; while an elementary understanding of the theorem’s implications may indeed be attainable, a rigorous engagement with its proof necessitates substantial familiarity with algebraic and diophantine concepts, including the descent method, abelian group structures, and the arithmetic geometry of height functions. mordell and weil’s finite generation theorem, thus, implicates not merely the boundedness of rational points but also exemplifies the structural richness and the intrinsic limitations that these elliptic curves exhibit within the broader mathematical landscape, solidifying its importance within the annals of number theory and underscoring its enduring significance in the study of elliptic structures over the rational field 130.74.58.21 (talk) 23:48, 14 November 2024 (UTC)

Wow, thanks very much for the detailed response. I understood a fair amount of it and will try to digest it some more. I think I'm still confused on a fairly basic issue and will try to figure out what I'm missing. The issue is that we are talking about a finite group, right? So can we literally write out the whole group table and find the subgroup structure? That would be purely combinatorial so I must be missing something. 2601:644:8581:75B0:0:0:0:2CDE (talk) 03:25, 15 November 2024 (UTC)

Oh wait, I think I see where I got confused. These are elliptic curves over Q rather than over a finite field, and the number of rational points is usually infinite. Oops. 2601:644:8581:75B0:0:0:0:2CDE (talk) 10:09, 15 November 2024 (UTC)

This response is pretty obviously LLM-generated, so don't expect it to be correct about any statements of fact. 100.36.106.199 (talk) 18:26, 15 November 2024 (UTC)

Yeah you are probably right, I sort of wondered about the verbosity and I noticed a few errors that looked like minor slip-ups but could have been LLM hallucination. But, it was actually helpful anyway. I made a dumb error thinking that the curve group was finite. I had spent some time implementing EC arithmetic on finite fields and it somehow stayed with me, like an LLM hallucination.

I'm still confused about where GRH comes in. Like could it be that rank E = 29 if GRH, but maybe it's 31 otherwise, or something like that? Unfortunately the question is too elementary for Mathoverflow, and I don't use Stackexchange or Reddit these days. 2601:644:8581:75B0:0:0:0:2CDE (talk) 22:32, 15 November 2024 (UTC)

Ok so I don't know anything about this but: it seems that the GRH implies bounds of various explicit kinds on various quantities (e.g.) and therefore you can end up in a situation where you show by one method that there are 29 independent points, and then also the GRH implies that the rank is at most 29, so you get equality. There is actually some relevant MO discussion: [2]. Here is the paper that used the GRH to get the upper bound 28 on the earlier example. 100.36.106.199 (talk) 23:55, 15 November 2024 (UTC)

Thanks, I'll look at those links. But, I was also wondering if there is a known upper bound under the negation of the GRH. 2601:644:8581:75B0:0:0:0:2CDE (talk) 02:47, 16 November 2024 (UTC)

Yeah I don't know anything about that, but it seems like a perfectly reasonable MO question. 100.36.106.199 (talk) 02:14, 20 November 2024 (UTC)

November 15

Are there morphisms when enlarging a prime field sharing a common suborder/subgroup ?

Simple question : I have a prime field having modulus $p$ where p−1 contains $O$ as prime factor, and I have a larger prime field $q$ also having $O$ as it’s suborder/subgroup. Are there special cases where it’s possible to lift 2 $p$’s elements to modulus $q$ while keeping their discrete logarithm if those 2 elements lies only within the $O$’s subgroup ? Without solving the discrete logarithm of course ! 82.66.26.199 (talk) 11:36, 15 November 2024 (UTC)

Clearly it is possible, since any two groups of order o are isomorphic. Existence of a general algorithm, however, is equivalent to solving the discrete log problem (consider the problem of determining a non-trivial character). Tito Omburo (talk) 11:40, 15 November 2024 (UTC)

So how to do it without solving the discrete logarithm ? Because of course, I was meaning without solving the discrete logarithm. 2A01:E0A:401:A7C0:9CB:33F3:E8EB:8A5D (talk) 12:51, 15 November 2024 (UTC)

It can't. You're basically asking if there is some canonical isomorphism between two groups of order O, and there just isn't one. Tito Omburo (talk) 15:00, 15 November 2024 (UTC)

Even if it’s about enlarging instead of shrinking ? Is in theory impossible to build a relation/map or is that no such relation exists yet ? 2A01:E0A:401:A7C0:9CB:33F3:E8EB:8A5D (talk) 08:48, 16 November 2024 (UTC)

At least into the group of complex roots of unity, where a logarithm is known, it is easily seen to be equivalent to discrete logarithm. In general, there is no relation between the groups of units in GF(p) and GF(q) for p and q distinct primes. Any accidental isomorphisms between subgroups are not canonical. Tito Omburo (talk) 15:02, 16 November 2024 (UTC)

November 16

What’s the secp256k1 elliptic curve’s rank ?

Simple question : what’s the rank of secp256k1 ?
I failed to find how compute the rank of an elliptic curve using the version of online tools like SageMath or Pari/gp since it’s the only thing I have access to… 2A01:E0A:401:A7C0:9CB:33F3:E8EB:8A5D (talk) 15:44, 16 November 2024 (UTC)

I don't know a clear answer but a related question is discussed here. 2601:644:8581:75B0:0:0:0:2CDE (talk) 01:57, 17 November 2024 (UTC)

Although I know it doesn t normally apply to this curvd, I was reading this paper https://pdfupload.io/docs/4ef85049. As a result, I am very curious about knowing the rank of secp256k1 which is why I asked it especially if it allows me know how to compute them on ordinary curves. 2A01:E0A:401:A7C0:417A:1147:400C:C498 (talk) 11:01, 17 November 2024 (UTC)

Maybe by some chance, this might have the answer. ExclusiveEditor _{Notify Me!} 19:20, 17 November 2024 (UTC)

Same question by same questioner, so not by chance.  --Lambiam 06:51, 18 November 2024 (UTC)

Yes, It’s me who asked the question. He didn’t replied to my last comment about the elliptic curve prime case. I’m meaning the paper 2A01:E0A:401:A7C0:9CB:33F3:E8EB:8A5D (talk) 07:08, 18 November 2024 (UTC)

November 17

Final four vote probability

In a social deduction game at the final four where nobody is immune and each of the four gets one vote what is the probability of a 1–1–1–1 vote? (78.18.160.168 (talk) 22:26, 17 November 2024 (UTC))

Social deduction games exist in many different versions, with different rules. Can you provide (a link to) a description of the precise rules of the version of the game you want us to consider?

Moreover, if the players can follow different strategies, or can follow their intuitions instead of rolling the dice and using the outcome according to the fixed strategy, the situation cannot be viewed as a probability problem. Can we assume that the players play with the same given independent and identically random strategy?  --Lambiam 06:47, 18 November 2024 (UTC)

I was thinking of The Traitors, but it could also be applied to Survivor: Pearl Islands. There are no dice. In The Traitors before the final four banishment vote, there is a vote on whether to end game or banish again. If everyone votes to end the game the game ends but if one or more people votes to banish again, the game continues. I jumped ahead to the banishment vote because I have not seen a season where all four people vote to end the game. PS my IP address has changed. (78.16.255.186 (talk) 20:24, 18 November 2024 (UTC))

I don't understand the rules from the description in The Traitors and don't know what a "1" vote signifies, but in any case, this does not look like it can be modelled as a mathematical probability problem, for a host of reasons. The outcome of a vote will generally depend on the dispositions of the participants (are they more rational or more likely to choose on a whim; are they good in interpreting the behaviour of others) as well on their past behaviours. It is not possible to assign probabilities to such factors, and there is no mathematical model for how such factors influence the voting.  --Lambiam 03:58, 19 November 2024 (UTC)

If you simplify much further to just "if you have four people, and each one randomly chooses someone (that is not the person themself), what's the probability that each person gets chosen once", then we can generalize this to some arbitrary $n$ people.

Let us assign each person some number from $1$ to $n$, so that each choice can be thought of as a mapping from $[[1,n]]$ to itself. When each person is chosen exactly once, this corresponds to a mapping from $[[1,n]]$ to itself where no number is mapped to itself. This is a derangement, and we can see that the number of ways of tied voting is exactly the number of derangements for $n$ people. Thus, the probability for $n$ is the number of derangements divided by the number of mappings where no one votes for themselves.

The number of derangements on $n$ elements is the subfactorial of $n$, denoted $!n$. As for total number of mappings, each of the $n$ people has $n-1$ choices, so there are $(n-1)^{n}$ such mappings. This brings the probability to $!n/(n-1)^{n}$.

For $n=4$ the number of derangements is $9$, and there are $81$ mappings where no one votes for themselves, so the probability is $1/9$. More generally, $!n=round(n!/e)$, so the probability in general is ${\frac {round(n!/e)}{(n-1)^{n}}}$. Note that this tends to $0$ as $n$ increases. GalacticShoe (talk) 06:00, 19 November 2024 (UTC)

November 19

Basic equations / functions in predicting probability of success in insurgent vs. conventional military engagements in mid-to-late 20th century warfare / calculations for probability of the success of insurgent movements (esp. with consideration of intangible factors)

can someone kindly uncover casualty rolls -
I am thinking in particular about the Ukrainian Insurgent Army and the debates which went on within the American special services in the late 40s through early 50s about providing assistance to them after the breakthrough of the 'Iron company' (you can look up on ukr, pol, rus wiki about the so-called Iron company of the UPA ; Залiзна сотнья) from Transcarpathia in Communist-occupied Ukraine through Czechoslovakia through to Bavaria (where there were already in residence many leaders of the Ukrainian movement who had been interned by the Germans, most prominent among these Stepan Andriiovich, of course, working to raise the Ukrainian issue in the consciousness both of the public in Western 'free' world, and in the minds of the military-political authorities, who were still reeling from the taste in their mouths of the 'betrayal' of Poland, which Churchill railed against, closer, as he was, to the heart of the issue, if we have these figures, we can make at the very least basic calculations, and predict with a degree of accuracy, for example, based on the help that the Americans were considering to render to the Ukrainian freedom fighters, the successes which they could have achieved considering also the concurrent armed struggles in Romania, in Poland, in the Baltic states — Preceding unsigned comment added by 130.74.59.208 (talk) 15:15, 18 November 2024 (UTC)

This all seems very interesting, but I don't see it as mathematics question. I suggest you try the History Stack Exchange. --RDBury (talk) 19:19, 18 November 2024 (UTC)

i should like to refuse with one regard only the question pertains to application of mathematics and hard sciences in interpretation of historical events and possibilities 130.74.59.186 (talk) 20:02, 18 November 2024 (UTC)

Full stops were invented for a reason: they are very useful in making text understandable.  --Lambiam 04:07, 19 November 2024 (UTC)

There is no mathematical theory that can be used for determining the probabilities of the possible outcomes of a real-world conflict. It is not even clear that the notion of probability applies in such situations.  --Lambiam 04:14, 19 November 2024 (UTC)

This seems like more the province of game theory than probability. That it's modelled using probability in e.g. simulations, such as computer games or board games, is due to the limitations of their models. They can't fully model the behaviour of all actors so they add random probabilistic factors to compensate. But those actually engaged in conflict aren't going to be using randomness, just the best strategy based on what they know about the conflict, including what the other side(s) will do. That's game theory.--217.23.224.20 (talk) 15:49, 19 November 2024 (UTC)

November 20

Sequences: Is there a name for a sequence, all of whose members are different from each other?

2A06:C701:7455:4600:C907:E8C0:F042:F072 (talk) 09:07, 20 November 2024 (UTC)

A term used in the literature: injective sequence.^[3]  --Lambiam 13:18, 20 November 2024 (UTC)

November 21

Is it possible to adapt Nigel’s Smart algorithm for establshing an isomorphism when the curve is only partially anomalous ?

An anomalous elliptic curve is a curve for which $\#E(\mathbf {F} _{q})=q$. But in my case, the curve has order j×q and the underlying field has order i×q. In the situation I’m thinking about, I do have 2 points such as both G∈q and P∈q subgroup and where P=s×G.

So since the scalar $s$ lies in a common part of the additive group from both the curve along it’s underlying base field, is it possible to transfer the discrete logarithm to the underlying finite field ? Or does anomalous curves requires the whole embedding field’s order to match the one of the curve even if the discrete logarithm solution lies into a common smaller group ?

If yes, how to adapt the Nigel’s smart algorithm used for solving the discrete logarithm inside anomalous curves ? The aim is to etablish an isomorphism between the common subgroup generated by E and $F_{p}$ 82.66.26.199 (talk) 19:47, 21 November 2024 (UTC)

November 22

Dihedral primes in base 36

Fourteen-segment display (alphanumeric display) can be used in base 36 (the largest case-insensitive alphanumeric numeral system using ASCII characters), thus we can use fourteen-segment display to define dihedral primes in base 36 (with A=10, B=11, C=12, …, Z=35), just like seven-segment display to define dihedral primes in base 10. If we use fourteen-segment display to define dihedral primes in base 36 (with A=10, B=11, C=12, …, Z=35), which numbers will be the dihedral primes in base 36 with <= 6 digits? 218.187.66.155 (talk) 19:14, 22 November 2024 (UTC)

It depends on how you encode each symbol on a fourteen-segment display (in particular, the number 0 and the letter O will need to be distinguished). If we go by File:Arabic number on a 14 segement display.gif and File:Latin alphabet on a 14 segement display.gif, then there are ten valid inversions, which are as follows: 0 <-> 0, 2 <-> 5, 8 <-> 8, H (17) <-> H, I (18) <-> I, M (22) <-> W (32), N (23) <-> N, O (24) <-> O, X (33) <-> X, and Z (35) <-> Z. Of these, only 5, H, N, and Z are coprime to 36, so any dihedral prime must necessarily end with one of these. Duckmather (talk) 04:02, 25 November 2024 (UTC)

November 23

radial distance between a circle and another enclosing circle

On an x-y plane, draw a circle, radius r1 centered on the origin, 0,0. Draw a second circle centered on some offset value -x, y = 0, radius r2 which greater than r1+x so that the second circle completely encloses the first and does not touch it. Draw a line at angle a beginning at the origin and crossing both circles. How do I calculate the distance along this line between the two circles? ```` Dionne Court (talk) 06:07, 23 November 2024 (UTC)

Given:

• inner circle: centre at $(0,0),$ radius $r_{1},$ equation $x^{2}+y^{2}=r_{1}^{2};$
• outer circle: centre at $(u,0),$ radius $r_{2}>r_{1}+u,$ equation $(x-u)^{2}+y^{2}=r_{2}^{2}.$
• line through origin at angle $\alpha ,$ parametric equation $(x,y)=(\lambda \cos \alpha ,\lambda \sin \alpha ).$

The line crosses the inner circle at $(x,y)=(\pm r_{1}\cos \alpha ,\pm r_{1}\sin \alpha ),$ both obviously at distance $r_{1}$ from the origin.

To find its crossings with the outer circle, we substitute the rhs of the line's equation for $(x,y)$ into the equation of the outer circle, giving $(\lambda \cos \alpha -u)^{2}+(\lambda \sin \alpha )^{2}=r_{2}^{2}.$ We need to solve this for the unknown $\lambda$. This is a quadratic equation; call its roots $\lambda _{1}$ and $\lambda _{2}.$ The corresponding points are at distances $|\lambda _{1}|$ and $|\lambda _{2}|$ from the origin.

The crossing distances are then $|\lambda _{1}|-r_{1}$ and $|\lambda _{2}|-r_{1}.$

If you use $\left|(|\lambda _{1}|-r_{1})\right|$ and $\left|(|\lambda _{1}|-r_{1})\right|,$ this will work for any second circle, also of it intersects the origin-centred circle or is wholly inside, provided the quadratic equation has real-valued roots.  --Lambiam 08:46, 23 November 2024 (UTC)