Even without knowing about weak and strong forces, Einstein had ideas to unify at least gravity and electromagnetism - a theory of everything (TOE). One idea is that matter itself is created by a state of spacetime. His way of thinking was ahead of his time.
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an unsolvable problem? and an unsolvable problem II.
Here more examples: for q=37 and p=31 there are several exceptions of different lengths. The expectation value L=30 can be produced alternatively and exceeded with lengths L=31 und L=32 in different ways. The occurrence of these structures is surprising.
| q | L | divisibility chain |
u |
|---|---|---|---|
| 29 | 22 | (u,21,19,17,15,13,11,3,7,5,3,23,29,3,5,7,3,11,13,15,17,19,21,v) |
? |
| 31 | 28 | (u,3,5,23,21,19,17,15,13,11,3,7,5,3,31,29,3,5,7,3,11,13,15,17,19,21,23,5,3,v) |
? |
| 37 | 30 | (u,29,3,5,23,21,19,17,15,13,11,3,7,5,3,31,37,3,5,7,3,11,13,15,17,19,21,23,5,3,29,v) |
? |
| 37 | 30 | (u,23,15,37,7,3,13,5,3,19,11,21,5,17,3,31,29,15,7,13,3,11,5,3,23,7,3,5,19,3,17,v) |
? |
| 37 | 31 | (u,15,7,11,3,13,5,3,19,7,3,5,17,3,11,[29],105,23,13,3,[31],5,3,7,[37],33,5, 19,3,17,7,15 13,v) |
? |
| 37 | 32 | (u,3,35,17,3,19,23,15,13,7,3,[29],5,3,[31],11,21,5,[37],3,17,13,15,7,19,3,11,5,3,23,7,3,5,v) |
? |
| 37 | 32 | (u,21,5,23,3,13,17,15,7,11,3,19,5,3,[29],7,3,5,13,3,11,[31],105,17,[37],3,23,5,3,7,19,3 11 13,5,v) |
? |
to be continued...
Recently in this blog: an unsolvable problem? This time the biggest prime number is denoted as q, and p is reserved for the second largest prime number. Here some solutions in the table: for a largest prime number q=7 (p=5) und u=47, u+1 is divisible by 3, u+2 is divisible by 7 etc., or for q=11 (p=7) and u=514 the number u+1 is divisible by 5, u+2 is divisible by 3 etc., and u und v are both not divisible by any of the given primes (u means undivisible or unit). You can recognize a construction pattern, which produces chains of length L=p-1. This cannot be proven, because there are exceptions or anomalies. For q=23 and p=19 there is an anomalous solution of chain length L=19 given in the table. The values of u can be computed with the Chinese remainder theorem.
| q | L | divisibility chain |
u |
|---|---|---|---|
| 7 | 4 | (u,3,7,5,3,v) |
47 |
| 11 | 6 | (u,5,3,11,7,3,5,v) |
514 |
| 13 | 10 | (u,3,7,5,3,11,13,3,5,7,3,v) |
? |
| 17 | 12 | (u,11,3,7,5,3,17,13,3,5,7,3,11,v) |
? |
| 19 | 16 | (u,15,13,11,3,7,5,3,17,19,3,5,7,3,11,13,15,v) |
? |
| 23 | 18 | (u,17,15,13,11,3,7,5,3,23,19,3,5,7,3,11,13,15,17,v) |
? |
| 23 | 19 | (u,3,13,5,3,7,11,3,5,19,3,23,7,15,17,13,3,11,5,21,v) |
? |
to be continued...
This video is a kind of explanation of what a spinor is together with an animation of the strange rotational behaviour. It takes a 720° rotation to bring it back to its original state. The mathematics of spinors is not yet understood completely but they have been important tools in modern physics since the invention of the Dirac equation (which was almost hundred years ago).
Update 2024-11-17: the video has been removed by the uploader
Norman Wildberger proposes a different approach than set theory as the foundation of mathematics. Can a different point of view lead to new insights?
Allen Hatcher published the book Algebraic Topology via Cambridge University Press. An electronic version is available for noncommercial personal use.
Algebraic Topology
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PDF incl. corrections, but no clickable Table of Contents
The following problem looks like a riddle. But careful: it's quite difficult and resists all approaches.
Take some prime numbers, e.g. M={11,41,61}, and search for a chain of numbers such that each is divisible by at least one of the chosen prime numbers, e.g. 121,122,123 (11 × 11, 61 × 2, 41 × 3). In this example the chain length is 3. Since the smallest prime number is bigger than the number 3 of chosen primes, the length of the chain is bounded by the number of primes. No prime can be used more than once.
There is a special role for prime number 2 thus it is forbidden. We choose a maximal prime number p und collect all odd primes up to p inluded in a set M, e.g. p=7 and M={3,5,7}. Here yo can find the chain 48,49,50,51 of length 4. Divisibility by 3 was used twice. Thus the chain gets longer than the number of primes: 4>3.
Or choose p=11 and M={3,5,7,11}. We find the chain 515,516,517,518,519,520 with length 6. The prime numbers 3 and 5 are used twice each. Thus the chain is with 6>4 longer than the number of primes.
The set M is determined uniquely by the maximal prime number p.
Question: how long is the maximum length of a chain to a given maximum prime number p (and therefore given set M), with each number in the chain divisible by at least one prime number in M?
For an explicitely given prime number p we can try, because the pattern will repeat after a finite number. This is because of modular arithmetic. We can then make a guess, because there is a pattern. Is it possible to proof the arising conjecture?
WP: Modular arithmetic
WP: Chinese remainder theorem
WP: Cyclic group
WP: Unit (ring theory)
WP: Multiplicative group of integers modulo n
Allen Hatcher published the book Topology of Numbers via AMS bookstore. It is a book about algebraic number theory, quadratic forms and quadratic fields. There is a PDF version available as download from his web page at Cornell University.
Topology of Numbers
PDF minor revision
PDF print version
AMS bookstore
Herbert S. Wilf published the book generatingfunctionology about generating functions, which is an interesting introduction to analytic number theory. The 2nd edition from 1994 can be downloaded legally for educational but not commercial purposes. Please read the copyright notice on the page.
The book "generatingfunctionology" incl. copyright notice
PDF classic
PDF with internal hyperlinks