2022 - A Good Year for Mathematics

For this post, I refer readers to the summary by Konstantin Kakaes of Quanta Magazine: The Biggest Math Breakthroughs in 2022 | Quanta Magazine.

Also see this video version: https://youtu.be/Nmgl78a02ys

 

Multiple Dimensions of Time - Part 3

                                            

Credit: Pixabay/CC0 Public Domain

In two prior posts, I explored how different authors and physicists have considered the possibility of multiple dimensions of time. See: Multiple Dimensions of Time and Multiple Dimensions of Time - Part 2.

As unusual as this topic may seem, I'm reading more about how many scientists have considered multiple dimensions of time to be a possibility. Just recently, another article was published this past week in which authors Andrzej Dragan et al. develop an extension of special relativity in 1+3 dimensional spacetime to account for superluminal inertial observers and show that such an extension rules out the conventional dynamics of mechanical point-like particles and forces one to use field-theoretic framework.

Update 12/25/2022 - another article posted: https://mindmatters.ai/2022/12/did-physicists-open-a-portal-to-extra-time-dimension-as-claimed-2/

A Million Dollar Prize for Solving a Math Problem

 

(Yitang Zhang, Photo credit: http://www.voachinese.com/media/video/i-america-math-zhang-yitang-20131204/1803128.html)

The Riemann hypothesis, first proposed by Bernhard Riemann in 1859, deals with the distribution of prime numbers. At the beginning of the 20th century, it was one of the top 23 problems identified by David Hilbert for the mathematical community to work on for the upcoming century. Math Vacation: David Hilbert Problems (jamesmacmath.blogspot.com) Riemann hypothesis - Wikipedia

The Riemann hypothesis and five others remain unsolved. The Clay Mathematics Institute has offered a US$1 million prize to anyone who can solve the Riemann hypothesis. Millennium Problems | Clay Mathematics Institute

The list of unsolved Hilbert problems may be closer to being shorten by University of California – Santa Barbara professor Yitang Zhang. https://www.scmp.com/news/china/science/article/3198779/has-chinese-born-professor-discovered-big-piece-150-year-old-maths-puzzle

If successful, this would be remarkable for a 67-year-old mathematician. However, it won’t be the first time for Zhang to solve a difficult problem later in life. Nine years ago, he shocked the world with his work of the twin-prime conjecture. While he didn’t prove the twin prime conjecture, he did prove there exists a limit below which (at the time it was 70 million) there must be an infinite number of primes separated by that a specific gap size, N. Since then, using Zhang’s techniques, the limit for that minimum gap size has been dropped from 70 million to a few hundred. If mathematicians could bring this limit down to 2, it would prove the twin prime conjecture.

August 20, 2023 Update: New York Post Article - Riemann hypothesis: Unsolved math problem worth $1 million (nypost.com).

Multiple Dimensions of Time - Part 2

 

(Graphic: Kalash on Iconfinder)

In a recent post, I explored how different authors and physicists have considered the possibility of multiple dimensions of time. Nature just published an article, Dynamical topological phase realized in a trapped-ion quantum simulator | Nature, in which an experiment using laser-emitted pulses sent in a Fibonacci sequence at 10 ytterbium qubits made the system behave as if there are two distinct directions of time. 

Additional links regarding this experiment:

Scientists Fed the Fibonacci Sequence Into a Quantum Computer and Something Strange Happened (futurism.com)

Physicists Got a Quantum Computer to Work by Blasting It With the Fibonacci Sequence (gizmodo.com)

Readers of this Blog Earned $837 Risk Free

 

(Graphic: Laura Reen on Iconfinder)

Last year, this blog suggested buying United States I Bonds as way to protect savings from inflation. The I Bond series pays interest rates adjusted twice yearly for inflation. Last November, the interest rate was set at 7.12% and in April this year, the rate was adjusted to a 40-year high of 9.62%. Readers who took the advice of this blog and bought the maximum allowed amount of $10,000 earned $837 over the past year. Next month, the rate will be adjusted again. It is anticipated that it will likely be 2 – 3 % lower than the 9.62% rate I Bonds have been earning since April.

 Individual - Buying Series I Savings Bonds (treasurydirect.gov)

For information on the rates, Go to: https://www.treasurydirect.gov/indiv/research/indepth/ibonds/res_ibonds.htm#irate

If you don't have an account already, set one up here: https://www.treasurydirect.gov/RS/UN-AccountCreate.do

Multiple Dimensions of Time

 

(Graphic: Carbon Design on Iconfinder)

I’m currently reading a Robert Heinlein book, The Pursuit of the Pankera (The Pursuit of the Pankera | Arc Manor Books), in which a group of time-space travelers devised a machine to travel through parallel universes. The key to their machine is they discover that there are three dimensions of space and three dimensions of time. The book’s theme led me to look up how theorists have speculated that there may be more than one dimension of time.

There turns out to be quite several theories about multiple dimensions of time. Enough so there is a Wikipedia page dedicated to the topic: Multiple time dimensions - Wikipedia.

Some of theories include compact time dimensions analogous to the spatial dimensions of string theory, two time dimensions in which one is in real time and one is in imaginary time as with complex numbers, and multiple time dimensions similar to conventional time.

Beyond Heinlein, other popular authors have employed multiple time dimensions including C. S. Lewis (https://en.wikipedia.org/wiki/Chronicles_of_Narnia) and J. R. R. Tolkien (https://en.wikipedia.org/wiki/The_Lord_of_the_Rings).

NASA Smashes an Asteroid (DART Mission)

 

(Image: Iconfinder.com)

NASA recently completed the experiment of colliding a spacecraft into an asteroid. The mission’s name was DART (Double Asteroid Redirection Test) and had the goal of determining if a small asteroid could be deflected. Ultimately, if the experiment worked, mankind would have a possible tool to change the trajectory of an Earth-bound asteroid, therefore avoiding a extinction event.

Remembering my high-school physics, the change in the velocity (speed and direction) could be determined by conservation of energy and conservation of momentum equations. I was going to try to re-create the needed equations, when I found that Rhett Allain (in WIRED.com) completed this exercise and explains the some of the nuances associated with calculating the change of the velocity of the impacted asteroid. Link: The Physics of Smashing a Spacecraft Into an Asteroid.

The calculations showed the target asteroid could have its velocity changed by about 1mm/sec. That doesn't seem like much, but if an incoming asteroid could be intercepted with enough time before impact, that small change could make enough of a difference to avoid impact with the Earth.

Update 10/11/2022 - NASA confirmed that they have detected a measurable difference in the asteroid's velocity: NASA confirms humans changed the motion of a celestial object for the first time (msn.com)

NASA says asteroid mission was successful, altered orbit by 32 minutes (msn.com)

Another post gives a method for calculating the probability of extinction events: Math Vacation: The Doomsday Calculation (jamesmacmath.blogspot.com)

Exploring Ending Digits of Prime Numbers in Other Bases

(Image:  Phosphor Light icons - Iconfinder)

I've been reviewing the distribution of ending digits of prime numbers expressed in base 10 and in other bases. For the purpose of this exercise, I used the first 304 prime numbers which begin with 2 and end with 2003. This list is at bottom of this post. 

Starting with base 10 is a good starting point. Most people have seen that these primes only end with digits 1, 2, 3, 5, 7 and 9. After 5, 2 and 5 are no longer the ending digit of primes since even numbers greater than 2 are not prime and multiples of 5 all end in 5 or 0. Excluding 2 and 5, the distribution of ending digits of primes in from this list is as follows:

Ending    Distribution
Digit

1                73
3                79
7                77
9                73

Tables of these first 304 primes expressed in bases 2 through 23 and the distribution of the prime number ending digits in these bases is available here.

Here is excerpt from the table showing the distribution for bases 2-10:

			Bases 2-10 distribution of ending digits (first 304 primes)
Ending Digit	2	3	4	5	6	7	8	9	10
1	303	148	147	73	148	46	68	47	73
2	0	155	1	78	1	48	1	51	1
3	0	0	156	79	1	53	78	1	79
4	0	0	0	73	0	50	0	51	0
5	0	0	0	0	154	54	79	54	1
6	0	0	0	0	0	52	0	0	0
7	0	0	0	0	0	0	78	50	77
8	0	0	0	0	0	0	0	50	0
9	0	0	0	0	0	0	0	0	73
0	1	1	0	1	0	1	0	0	0

Some observations from the above table:

In base 2, all primes end with a 1 except for 2 in base 2 which is 10 and it ends in 0. This makes sense because all primes greater than 2 are odd numbers and all odd numbers in base 2 end with the digit 1.

Notice that all primes over 3 in base 6 end in the digits 1 or 5. This can be explained by the fact that all primes over 3 are multiples of 6 minus 1 or plus 1 (for example 5 and 7). See these two other posts for more information on this prime number property:

A Prime Number Property Rediscovered 

Prime Number Sandwich

Base 7 is also interesting in its distribution of ending digits in that it includes all possible digits 0 through 6. The prime 7 expressed in base 7 is 10; however, that is the last time in this base that a prime ends in 0 because any higher number in base 7 ending in 0 will be a multiple of 7 (such as 14 in base 7 is 20 and 49 in base 7 is 100) and therefore, is not prime. Also, the ending digits are fairly even distributed between the digits 1 through 6.

The distribution for base 7 prime ending digits begs the question of whether there are bases higher than 10 for which all possible ending digits are used. The attached spreadsheet has a tab for bases 11 through 23. The summary of ending digit distributions is (note - letters are used as digits to express numbers in bases higher than 10):

				Bases 11 through 19 - Ending digit distribution (first 304 primes)
Ending digit	11	12	13	14	15	16	17	18	19	20	21	22	23
1	27	70	25	46	34	34	20	47	14	35	22	27	16
2	33	1	28	1	40	1	21	1	17	1	26	1	13
3	30	1	26	53	1	39	20	1	19	39	1	30	15
4	29	0	25	0	37	0	17	0	21	0	25	0	15
5	30	77	25	54	1	38	17	54	19	1	26	30	13
6	30	0	24	0	0	0	16	0	15	0	0	0	12
7	33	78	26	1	38	40	20	50	18	41	1	33	16
8	32	0	26	0	39	0	20	0	17	0	24	0	11
9	29	0	25	47	0	34	18	0	15	35	0	29	13
0	1	0	1	0	0	0	1	0	1	0	0	0	1
A	30	0	25	0	0	0	19	0	17	0	24	0	15
B	0	77	24	50	39	39	20	50	19	38	25	1	14
C	0	0	24	0	0	0	20	0	16	0	0	0	12
D	0	0	0	52	39	41	19	51	19	40	26	32	11
E	0	0	0	0	36	0	21	0	14	0	0	0	14
F	0	0	0	0	0	38	19	0	14	0	0	29	15
G	0	0	0	0	0	0	16	0	15	0	22	0	14
H	0	0	0	0	0	0	0	50	16	36	28	30	15
I	0	0	0	0	0	0	0	0	18	0	0	0	14
J	0	0	0	0	0	0	0	0	0	38	28	32	15
K	0	0	0	0	0	0	0	0	0	0	26	0	13
L	0	0	0	0	0	0	0	0	0	0	0	30	16

It appears that when the base is a prime, that all possible digits are found as the ending digit of prime numbers expressed in these other bases. I say "appears" because this is a limited table (limited in number of primes and in bases) and can't be considered a proof. The table shows this for prime bases up to 23. Although not included in the table, I've also confirmed this for primes up to 31.

In a related topic, Dirichlet's theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. I need to explore this theorem and the distribution of prime ending numbers to see how these concepts are linked.

This is another link discussing ending digits.

Numberphile has an interesting video about ending digits of prime numbers: (224) The Last Digit of Prime Numbers - Numberphile - YouTube

List of first 304 primes (2 through 2003) used for the distribution tables above:

2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
127
131
137
139
149
151
157
163
167
173
179
181
191
193
197
199
211
223
227
229
233
239
241
251
257
263
269
271
277
281
283
293
307
311
313
317
331
337
347
349
353
359
367
373
379
383
389
397
401
409
419
421
431
433
439
443
449
457
461
463
467
479
487
491
499
503
509
521
523
541
547
557
563
569
571
577
587
593
599
601
607
613
617
619
631
641
643
647
653
659
661
673
677
683
691
701
709
719
727
733
739
743
751
757
761
769
773
787
797
809
811
821
823
827
829
839
853
857
859
863
877
881
883
887
907
911
919
929
937
941
947
953
967
971
977
983
991
997
1009
1013
1019
1021
1031
1033
1039
1049
1051
1061
1063
1069
1087
1091
1093
1097
1103
1109
1117
1123
1129
1151
1153
1163
1171
1181
1187
1193
1201
1213
1217
1223
1229
1231
1237
1249
1259
1277
1279
1283
1289
1291
1297
1301
1303
1307
1319
1321
1327
1361
1367
1373
1381
1399
1409
1423
1427
1429
1433
1439
1447
1451
1453
1459
1471
1481
1483
1487
1489
1493
1499
1511
1523
1531
1543
1549
1553
1559
1567
1571
1579
1583
1597
1601
1607
1609
1613
1619
1621
1627
1637
1657
1663
1667
1669
1693
1697
1699
1709
1721
1723
1733
1741
1747
1753
1759
1777
1783
1787
1789
1801
1811
1823
1831
1847
1861
1867
1871
1873
1877
1879
1889
1901
1907
1913
1931
1933
1949
1951
1973
1979
1987
1993
1997
1999
2003