joigus

You see? Studiot was right. You're learning from your mistakes.

Who needs categories? It's not for a humble 'Beacon of Hope' to decide. You have my respect, that's all I can say. Plus you've got all the symptoms: a self-correcting mind, critical thinking, insatiable curiosity, relational thinking (pattern sniffing), imagination tempered by intellectual caution, no-nonsense. Are you fishing for compliments? That's another symptom: actors, sportspeople, and scientists share it.

You're quite right. In fact, it's not impossible to be both. And misunderstanding prowls everywhere.

I think this is very much a problem of categories, not as perfect classical (Aristotelian) categories (equivalence classes in maths), but as a Wittgenstein (family resemblance) kind of categories. A cat: A cat: Another cat: Not a cat:

I wouldn't be too surprised if there were something to those claims. During the Cold War there were similar episodes in science. Thank you.

Penitenziagite!

Do you mean understanding or misunderstanding?

1 is not a prime and those are not the first 23 primes. Don't trust me with numbers. The first 9 primes, I should have said. Thank you @studiot.

Rest in peace. Here's one of his many brilliant moments: "I suddenly remembered my Charlemagne: Let my armies be the rocks, and the trees, and the birds in the sky"

Even without much time (or energy) to think, it sounds about right to me. And the reason is that you're going to a point outside the light cone of both departure points. That is known technically as "outside the causal cone." The future light cone of event "departure from A" is seen as the future of "departure from A" by all inertial observers. Analogously respect to the past. But areas outside, like the point event you're proposing, are neither past nor future. Some inertial observers will see them as future of either A or B; others will see it as past.

I was a bit surprised that such a simple formula would hold for a problem like this. Prime numbers are quite unpredictable. Prime gaps for example cannot be predicted by a general formula. So a formula giving all possible sum decompositions of products of primes... I didn't see the flaw in the argument, because I didn't see the argument. But again, number theory is not within my comfort zone.

I see what you mean. But I think we basically agree about that: So the decomposition of 223'092'870 into a pair of numbers x+y with x, y, not necessarily being prime, but being relative primes of 2, 3, 5, ..., 23, I think is the defining condition. I'm way past my comfort zone here. Fortunately @Sensei did it for all of us, and even though I'm not a great code reader, I think it's what the OP was asking.

You guys are very valuable members of this community. I've been swept away by both of you several times. Yes, you're right, @studiot. But the OP was about the very special, very non-prime number \( 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23 \), factor of the first 9 prime numbers.

Why wouldn't rational and irrational numbers be summable?

Sorry, the problem reduces to how many numbers there are between 1 and 223'092'869 that are not divided by 1, 2, 3, 5, ..., 23.

Number 1 is a bit misleading, because \( \sqrt{1} = 1 \). Generally there are two basic ways of undoing a square root. One is squaring a root; e.g., \[\sqrt{a}=2\] which gives, \[a=4\] and the other is the one you suggest --rooting a square--, but with that one you must be careful: \[\sqrt{a^{2}}=4\] which gives, \[a=\pm4\] Another possible way to get square roots out of the way is to remember that sums times differences give differences of squares. As in, \[\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)=\sqrt{a}^{2}-\sqrt{b}^{2}=a-b\] You can prove quite amazing identities with this: \[\frac{\sqrt{a}}{\sqrt{a}+\sqrt{b}}+\frac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}=\frac{2a}{a-b}\] \[\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\frac{\sqrt{b}}{\sqrt{a}-\sqrt{b}}=-\frac{2b}{a-b}\] There's almost no end to fun with square roots! Exactly.

Now that I think about it, the product of the first 23 is, as you say, 223092870. Now, the number of ways in which you can sum two (arbitrary) natural numbers to give N is just 111'546'435, because 223'092'870 is even. So the problem reduces to finding how many numbers there are between 1 and 111'546'435 that are not divided by 1, 2, 3, 5, ..., 23 (the 1st 23 primes). The only thing I can say is that's not an elementary problem. How did you get your conjecture @Tinacity? Counterexample: \( \pi \) is irrational; \( 1-\pi \) is irrational too. but \( \pi + 1- \pi = 1 \).

She didn't specify, but I thought it was kind of implied... For arbitrary products of primes, certainly nothing like that can be proved with the present mathematics. But for 223092870, I just don't know.

The maths for predicting that kind of thing is called number theory. I know very little about number theory. It studies connections between numbers. The result that you propose reminds me of some theorems by Fermat. Have you proved it, or is it just an intuition? Maybe @wtf or @Sensei, or @mathematic, or @taeto can help you. @studiot is encyclopedic. Maybe he can help you too. Number theory is not very interesting for physics, AFAIK. And physics and mathematical physics are my turf.

It's a mark-up language; a subset of XML: https://www.w3.org/Math/whatIsMathML.html Sorry what I posted was inline LateX, but you can read it as MathML by right-clicking.

I will bee looking into this more closely later, and see if I can add anything significant, but first off I'd like to tell you about what seems to be a misconception in what you say in your question. Elementary particles are not like tinker-toy assemblies from which you can split the parts. They are instantiations of one basic thing, say the electron, of which you can obtain more and more copies by providing the necessary energy (2\( mc^2 \), with the electron mass.) By doing so, you produce particle-antiparticle pairs of the given kind. The universe before the big bang (following the standard inflationary model) had particle number zero. So, as others (Swansont, Bufofrog) say, no. This would not be the picture: Connecting with MigL's comments, all the fields must have to have been there (flavours, colours,...), but with zero expected value for the number of particles or antiparticles. This, in turn, connects with the very interesting question of "did the laws themselves appear at the moment previous to the bang, or were they already there?" Lee Smolin is one of the most notable proponents of this question. The electron-positron state that you talked about is called the positronium. It's very short lived. I don't think it lives long enough on average to produce relativistic velocities.

For all it's worth, I'll say users @Ghideon and @MigL are right AFAIK.

And yes, MigL is right. In order to get back to its original state you must turn it around 720º (4\( \pi \) radians). It was corroborated in experiments with external magnetic fields. Although that really may be confusing to you.

An ordinary rotation is cycling around an axis. An electron is not like that. No matter how you look at it, it always rotates with angular momentum h/2 or -h/2 in any direction you look. So no, it doesn't act like a planet. Electrons do not transform into photons; the either emit photons of absorb them. Electrons might absorb photons if there are photons there to be absorbed. Electrons radiate when they are accelerated, or when they are in an excited state. I don't really understand what behaviour you are picturing the magnetic field to inhibit.

This could be hard proof of tunneling for macroscopic objects...