What is Riemann's hypothesis and can it be explained in simple terms?

[Dazzlex]

Solve this

1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
3 1 2 2 1 1
? ? ? ? ? ? ? ?​

 

[bel]

13112221

 

[Esd]

Each line describes the previous line (how many of each symbol there were). After 13112221,
1113213211. I can't see there being any characters but 1,2,3 in any of the later lines, and therefore also any character repeating more than 3 times in a line (ex.1 1 1 1). Also, in the long run, "1" to have most frequent appearance of the 3 numbers.

 

[matticus]

you can't solve something unless there is a problem. you might as well have said:
solve this
"bear, racecar, 7,?".

 

[HallsofIvy]

While I agree with matticus, I think both bel and Esd have devined what Dazzlex intended. And, of course, it has nothing to do with mathematics.

 

[neutrino]

And yet, this thread is still sitting in the Number Theory section.

 

[matt grime]

There are some interesting questions about the 'say what you see' sequence. See the work of Conway (its inventor) for more. And it is maths, Halls, though not necessarily number theory.

 

[HallsofIvy]

I don't think you have understood what Conway is saying.

 

[matt grime]

Really? I don't? So the fact that JH Conway (a mathematician in a maths department; this doesn't automatically make it maths, of course - has Conway been quoted as saying 'and of course this is completely unrelated to the entirety of mathematics') has done some interesting mathematicalwork on this isn't sufficient for it to be maths? And Conway's constant isn't an interesting thing? What on Earth does something have to be to be considered maths, then Halls? Perhaps be in a tedious book on calculus taught to engineers?

Exercise: prove that nearly all say what you see sequences (or look and say) have the ratio digits of consecutive numbers converges to a constant, further show it is algebraic, and find its minimal polynomial.

 

[CRGreathouse]

solve this
"bear, racecar, 7,?".

Raccoon.

 

[matticus]

so close, it was actually badger.

 

[nesna]

Solve this

1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
3 1 2 2 1 1
? ? ? ? ? ? ? ?​

I saw this puzzle on the last page of the book titled "mathematical mysteries:the beauty and mathematics of numbers" by Calvin C Clawson. I broke my head over it for awhile. And eventually couldn't resist the temptation to check the solution. Though the solution was there, plain and bare, there was not a description for how he arrived at the given solution.

Im glad i saw it on this thread today.

 

[Wild Angel]

Solve this

1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
3 1 2 2 1 1
? ? ? ? ? ? ? ?​

could it be??

1 3 1 1 1 2 1

 

[ramsey2879]

could it be??

1 3 1 1 1 2 1

No Bel and Esd got the right answer: "one 3 one 1 two 2's and two 1's" is the description of the previous term. The answer is this description in "shorthand", i.e., "13112221". See the posts by Matt Grime for more background on this type of sequence.

 

[LukeD]

By the way, in general, this is called (decimal) run length encoding. While it may not have direct applications to most of number theory, algebra, or combinatorics (though clearly some) and likely no useful connections to any other field of mathematics, it is a very commonly used compression method (precisely because it is extremely simple) for fairly regular strings of words. Thus, it is both studied and used by computer scientists.

You have a base set, a set of words, and a subset of the natural numbers (could this be generalized to an ordered groupoid? I'm not sure) that you use to count them with.

In some cases, the encoding of your string isn't going to be unique, in some cases it is.

For instance, let's say your words are A, B, and C, and you're using the numbers 1-99
Then the string AAAAAAABBCCCCCBBBAAAA encodes to 7A2B5C3B4A, which is significantly shorter than the length of the original string. The nice thing about this encoded string is that when you look at each symbol you know immediately whether it's a word or a number by its position, so you can expand it very easily.

Let's say your words are instead 0:A 1:B 2:AB 3:AAB 4:AAAB
Then the string BAAABBAABAAAAABAABABBAAA encodes to 111411133023121130, though clearly this is not unique.

Furthermore, you can allow a lot of really weird things with your words, like letter that means "erase the previous symbol if one exists" or something like that.

The sequence that this topic was started about of course uses the the numbers 0-9 as words and the numbers 1-9 as counters. The interesting thing is that it's done in such a way so that you can keep applying the code to the result.

There are a lot of interesting problems with this type of encoding. For instance, given a set, finite number of words, a set bound on your counters, and a set of strings, how can you find the optimal set of words to encode the set, and how given such an encoding, how can you find the optimal encoding of each string?

It's also fairly easy to show (I have not done this. I have not even done more than heard this topic in passing, but the result is intuitive, and I've heard that it's not too difficult to show) that if the words need to be stored along with the encoding of the sets, that if an optimal set of words can always be found, encoding a second time cannot decrease the total number of symbols needed to store the encoding of the entire set of strings along with the words

----------

Err... anyway, my point is that, yes, this is real math

 

[Wild Angel]

Cool. I think I understand this sequence now. thanks

 

[Dazzlex]

Well done to everyone who understood the problem. I saw it in a book and it startled me at first but after a while it just looked so simple, so I just had to share it :)

 

[nesna]

Well done to everyone who understood the problem. I saw it in a book and it startled me at first but after a while it just looked so simple, so I just had to share it :)

mathematical mysteries:the beauty and mathematics of numbers" by Calvin C Clawson

 

[Sean Torrebadel]

This is deductive reasoning right? Is the answer 13112221?

 

[Sean Torrebadel]

This is the general pattern that I see, is that right?:

x x x x x x x 1
x x x x x x 1 1
x x x x x x 2 1
x x x x 1 2 1 1
x x 1 1 1 2 2 1
x x 3 1 2 2 1 1
1 3 1 1 2 2 2 1
1 3 3 1 1 2 1 1
1 3 1 1 1 2 2 1
3 3 3 1 2 2 1 1
3 3
3 3
1 3
1 3
1 3

 

[nesna]

This is the general pattern that I see, is that right?:

x x x x x x x 1
x x x x x x 1 1
x x x x x x 2 1
x x x x 1 2 1 1
x x 1 1 1 2 2 1
x x 3 1 2 2 1 1
1 3 1 1 2 2 2 1
1 3 3 1 1 2 1 1
1 3 1 1 1 2 2 1
3 3 3 1 2 2 1 1
3 3
3 3
1 3
1 3
1 3

It is right uptil the 7th line. After that it's wrong, I am afraid !

 

[Sean Torrebadel]

I know, I never continued it past that line, just showed the pattern.

 

[nesna]

After the seventh line,

1 3 1 1 2 2 2 1

1 1 1 3 2 1 3 2 1 1

3 1 1 3 1 2 1 1 1 3 1 2 2 1

1 3 2 1 1 3 1 1 1 2 3 1 1 3 1 1 2 1 1 1

... and goes on i believe

 

[ramsey2879]

This is the general pattern that I see, is that right?:

x x x x x x x 1
x x x x x x 1 1
x x x x x x 2 1
x x x x 1 2 1 1
x x 1 1 1 2 2 1
x x 3 1 2 2 1 1
1 3 1 1 2 2 2 1
1 3 3 1 1 2 1 1
1 3 1 1 1 2 2 1
3 3 3 1 2 2 1 1
3 3
3 3
1 3
1 3
1 3

Then you are indicating that each column follows a repetitive pattern. That is inductively correct but that is wrong according to the actual rule used to form the sequence.

 

[ramsey2879]

Then you are indicating that each column follows a repetitive pattern. That is inductively correct but that is wrong according to the actual rule used to form the sequence.

You can of course deduce that each column will eventually follow a pattern just as the decimal expansions of rational numbers all eventually follow a pattern. But the eventual pattern for column 4 is not 1122 it is 1223, the first 1 is not part of the repeating period.

 

[fleem]

Technically, there is no right answer. Of course, I think most people at least subconsciously presume in a case like this that the intended question is, "what is the next output of the simplest possible predictive algorithm that predicts the given lines". If, for example, I take the question at face value, then I have a right to defend a claim that the next line should be a smudge of dust from inside a bass fiddle.

Some of you may be wondering why I state the obvious. Well, I'm not sure its so obvious. Is there a way to prove that the algorithm given is the simplest? We haven't even a good definition of "simple" nor "complex", let alone a way of knowing we've arrived at the "simplest" algorithm. I suppose a definition of "simplest algorithm" would be something like "the algorithm with the fewest steps", but then we must define the complexity of each instruction in the algorithm. So we see that each instruction of an algorithm is an algorithm in itself, and if we follow that thought, we end up at some arbitrarily chosen set of postulates and can go no further. So then the question must be asked, "are the chosen postulates the BEST ones for simplifying the algorithm."

This is why I dislike seeing such questions used in anything other than fun (I realize fun is the purpose here, though)--because we might be punishing geniuses when we tell them they got it wrong.

 

[ramsey2879]

You can of course deduce that each column will eventually follow a pattern just as the decimal expansions of rational numbers all eventually follow a pattern. But the eventual pattern for column 4 is not 1122 it is 1223, the first 1 is not part of the repeating period.

Oddly enough I found that the length of each repeating cycle is always one of the following lengths 1,2, or 4. For simplicity I considered only cycles of length 4 by making the repeating cycle 12... = 1212.. etc. Then I change the index of those cycles that are the same as a previous cycle with the index changed (i.e. 2121... = 1212...) so that only the follow cycles need be considered. I list them below (all 24 possibilities are represented) along with the column in which they first appear.

1111 1
1112 6
1113 5
1122 10
1123 13
1132 9
1133 32
1212 2
1213 14
1222 40
1223 4
1232 12
1233 49
1313 164
1322 102
1323 35
1332 7
1333 15
2222 3
2223 18
2233 19
2323 37
2333 87
3333 185

 

[ramsey2879]

Oddly enough I found that the length of each repeating cycle is always one of the following lengths 1,2, or 4. For simplicity I considered only cycles of length 4 by making the repeating cycle 12... = 1212.. etc. Then I change the index of those cycles that are the same as a previous cycle with the index changed (i.e. 2121... = 1212...) so that only [24] cycles need be considered.

Now I tried various starting combations and found that the repeating cycle for each column is only dependent upon the ending number as long as there no repetions of the same digit more than 3 and no ending group of two 2's( a ending combination of ...ABB) where B = 2 as a A cycle with two columns of 2's appended at the end. For instance if the starting combination end in 1, say you start the say and see sequence of 1122331 or any other ending with 1 and containing no group of more then 3 of the same digit, the first occurance of the repeating pattern 2223 will always be in column 18 and all other columns will always eventually have the same repeating cycle. If the starting combination ends in 12, 222, 32, or 3, there are only 6 possible repeating cycles and the first column of 4 of them will be the same. My next goal is to find the first column if any in which a given repeating cycle from these 6 cycles will be the same regardless of the combinations of 1's, 2's, and 3's with no grouping of 4 or more of the same digit together. In the listing below of the 24 cycles the 3rd, 4th and 5th columns give the first column in which given cycle will appear for starting combinations ending in 1, 2, or 3.
S cycle ?1 ?2 ?3
1 2222 3 1 4
2 1111 1 2 2
3 2323 37 6 6
4 3333 185 7 1
5 1212 2 8 8
6 1313 164 22 22
7 1112 6
8 1113 5
9 1122 10
10 1123 13
11 1132 9
12 1133 32
13 1213 14
14 1222 40
15 1223 4
16 1232 12
17 1233 49
18 1322 102
19 1323 35
20 1332 7
21 1333 15
22 2223 18
23 2233 19
24 2333 87

 

[phoenixthoth]

And, of course, it has nothing to do with mathematics.

Huh?

If these numbers weren't quantitative, such as addresses of houses, in other words essentially just names, then I'd be inclined to agree. But there's counting here, patterns, patterns about counts, the possibility to generalize and discern other patterns... How can that of course have nothing to do with math?

 

[Graviton]

Riemanns hypothesis

can anyone explain to me thoroughly the riemanns hypothesis...i don't understand it.