The system of positive integers is an example of perpetual motion, a metaphysical perpetual motion system. The integers lose no heat because they are not physical, entropy fluctuates in the system which can be measured by the recurrence of prime numbers and the prime gap. Poincare Recurrence exhibits itself in twin primes. The system is Boltzmann time symmetric as well.

Let , n a positive integer. If k is prime (an odd prime, as k>2), show that n must be a power of two.


Let n be a positive integer but not a power of 2. Then it must have at least one prime factor that is not two, and thus must have at least one odd factor greater than 1. So let n=ab, with b>1 odd; we see 1≤a<n and 1<b≤n.

Now, recall that for integers x and y and odd positive integer m, that is divisible by the integer x+y. Letting y=1, we see .
Substituting and m=b (as b is odd), we see
,
and since 1≤a<n, , we see that k has a factor which is neither unity nor k itself. Thus, if n is not a power of 2, then k cannot be prime. So if k is prime, n is a power of 2.

The numbers are known as Fermat numbers. The first few Fermat numbers are:







Those Fermat numbers which are prime are called Fermat primes. So our proof was that every prime of the form is a Fermat prime. Of the above list of Fermat numbers, F₀, F₁, F₂, F₃, and F₄ are prime, but F₅=4,294,967,297=641×6,700,417 is not prime. In fact, F₀, F₁, F₂, F₃, and F₄ are the only known Fermat primes. However, as the Fermat numbers grow so rapidly, little is yet known about those for large n, and there are a number of open problems, including whether there are any other Fermat primes besides those listed above, and if there are infinitely many or finitely many Fermat primes (see here).

Concepts:

• Divisibility Rules
• Prime Numbers
• Factors
• Composite Numbers

Activities:

• Assessment:
  • Unit 1Quiz Divisibility

The Riemann hypothesis: The greatest unsolved problem in mathematics
QA246 .S23
511

Prime obsession: Bernhard Riemann and the greatest unsolved problem in mathematics
QA246 .D47
512.73

While learning anything in life, there come some boring things you should know how to do, because they are usually a good representation of some mechanics or work flows -did that make any sense at all?-

So, in boredom, I hereby present you a prime number sequencer and a Koch curve, great examples of simple data and pattern visualization and recursion respectively.

If you are learning how to code computer graphics and can’t do this stuff… you can’t say you’re on the right track until you do.

I hope my chaotic coding style comes in handy for someone

After seeing someone else’s poster I found myself laying out the set of non-zero prime numbers below 31 onto an illustrator canvas. I was hoping to spend some time laying out nice lines with the numbers, but I got distracted drawing with the numbers. Finally I just had to leave it. Almost all prime in the end.

Contents:

• Prime and Composite Numbers
• Factors
• Prime factorization
• LCM
• GCF
• Divisibility Rules
  

Activities:

• Assessment: Unit 2  Test

——————————-

Contents:(1ºB)

• Prime and Composite Numbers
• Divisibility Rules
• GCF
• LCM
• Word Problems
• Prime factorization

Activities:

• Warm-Up: Review previews done exercises and clarify doubts
• Independent Practice: Review worksheet about the previously studied concepts.

Contents:(1ºA)

• Prime and Composite Numbers
• Divisibility Rules
• GCF
• LCM
• Word Problems
• Prime factorization

Activities:

• Warm-Up: Review previews done exercises and clarify doubts
• Independent Practice: Review worksheet about the previously studied concepts.

———————-

Contents:

• Multiples
• LCM
  

Activities:

• Wrap-Up: Read word problem from the blog and answer question in groups following the roles given to each member of the group (cooperative learning)
• Teacher Input: Use power point presentation to explain least common multiples and how to find it as well as compare with Greatest Common Factors
• Independent Practice: Given a set of exercises in the blog.  Find the Least Common Multiple of each set of numbers.
• Wrap-Up: Check answers and clarify doubts.

Integra Links:

Oscar and Sonia went on a car ride at the Amusement Park.  It takes Sonia 4 minutes to go around the track while it takes Oscar 6 minutes.

If they both leave at the same time, how long will it take them to meet at the starting point?

Draw a table to figure out how long will it take each of them to compleate each track.

Let´s see how it works:

GCF-LCM

Find the LCM of the following pairs of numbers:

a. (64, 80)

b. (130, 150)

c. (135, 125)

d. (10, 130)

e. (140,220)

f. (2, 3, 5)

g. (2, 5, 10)

h. (4, 12, 25)

i. (3, 8. 18)

Test Today!

• Divisibility Rules
• Least Common Multiple
• Greatest Common Factor
• Prime and Composite Numbers
• Prime Factorization (Division and Factor Tree)
• Word problems

И снова заработал инстинкт… самонасыщения)) Мол если не сделаю – опять голод. А этот метод собаки Павлова мне подходит.
Теперь у меня есть адекватный скрипт чтобы вычислить простые числа и отправить его в грид, сделав параметрический JDL.

#!/bin/bash
UPPER_LIMIT=$1 # From command line.
let SPLIT=UPPER_LIMIT/2 .

Primes=( ” $(seq $UPPER_LIMIT) )

i=1
until (( ( i += 1 ) > SPLIT ))
do
if [[ -n $Primes[i] ]]
then
t=$i
until (( ( t += i ) > UPPER_LIMIT ))
do
Primes[t]=
done
fi
done

#printf -e “\e${Primes[*]}\e”

printf “%s\n%s\n” ${Primes[*]}

exit 0

Преимущество заключается в том, что скрипт не имеет границ, а ждёт пока ему зададут, до какого числа считать. Это значит, что не придётся создавать отдельный текст файл с параметрами, и в sh не надо будет нудно прописывать чтобы он оттуда их считывал.
Итого: мало файлов (скрипт+JDL), хитрое решение.
Отлично. Теперь бы ещё узнать как из своего виртуального каталога кинуть это к Марио в (на?) его LFC…
Боже, хочу стать учёной, чтобы всё время одной сидеть и думать про всякую хрень, чтобы меня не дёргали и не отвлекали) И время от времени ходить на концерты Раммштайн))

Contents:

• Prime and composite numbers

Activities:

• Warm-Up: Students answer the question: What are prime or composite numbers?
  
• Teacher Input: Using the link given in the Math Daily Blog the teacher explains how to determine whether a number is prime or composite.
  
• Guided Practice: Interactive activity to practice prime and composite numbers.
  
• Independent Practice: Page 12 Exercises 2-6
  
• Wrap-Up: Extension problems 8 &8
  

Integra Links:

What are prime or composite numbers?

Prime numbers practice: (follow the link)

Class work: Page 12, Exercises 1-6

Homework: Page 13, Exercises 7 & 8

What are prime or composite numbers?

Prime numbers practice: (follow the link)

Prime Numbers

Class work: Page 12, Exercises 1-6

Homework: Page 13, Exercises 7 & 8

]]> http://planningmathcap.wordpress.com/2009/10/17/thusday-22-october-2009/ Sat, 17 Oct 2009 21:12:08 +0000 Carmen Ana http://planningmathcap.wordpress.com/2009/10/17/thusday-22-october-2009/

Contents:

• Prime and composite numbers

Activities:

• Wrap-Up: Students answer the question: What are prime or composite numbers?
  
• Teacher Input: Using the link given in the Math Daily Blog the teacher explains how to determine whether a number is prime or composite.
  
• Guided Practice: Interactive activity to practice prime and composite numbers.
  
• Independent Practice: Page 12 Exercises 2-6
  
• Wrap-Up: Extension problems 8 &8
  

Integra Links:

What are prime or composite numbers?

Prime numbers practice: (follow the link)

Class work: Page 12, Exercises 1-6

Homework: Page 13, Exercises 7 & 8

This two-minute video  shows how to multiply 21 x 13, and  123 x 321, with an easy line-drawing and angle-counting method.  No times tables needed!

Here is another very clear example showing 432 x 312, done with the same method.  However it shows a way to “carry” digits.

Seeing these videos makes me wonder anew whether math is really about “shapes,” and about describing all the “shapes” in the universe.

If there is anyone from China reading this blog, I’d really appreciate knowing how extensively this method is actually used in schools.

–Eileen

PROBLEMS

1.) Find a linear function such that and .

2.) Solve for :

3.) Prove that the product of consecutive numbers is always divisible by .

4.) Prove that if is prime, and are integers, and , then .

SOLUTIONS AND PROOFS

Post Date: October 20, 2009

1. Solution: This is just the same as saying, find the equation of the line passing through and . So, by point slope formula, we have, 

2.) Solution:

3.) Proof: A number is divisible by if it is divisible by and . A product of consecutive numbers is divisible by because at least one of them is even, so it remains to show it is divisible by .

If a number is divided by , its possible remainders are and .  Assume and be the three consecutive numbers, and be the remainder if is divided by .

Case 1: If , we are done.

Case 2: If , then

Case 3: If , then .

Since the product of the three consecutive numbers is even, and for each case of , one of the consecutive numbers is divisible by , the product of three consecutive numbers is divisible by

4.) Proof: From definition, for some

Raising both sides of the equation to , we have By the binomial theorem,  .

Notice that every term aside from is divisible by . (Why?). Therefore,  

Hence, then

OK.

1. I have a flat!

2… with no internet connection yet.

3. But the plums here are awesome!

4. The fountain in the Schillerplatz in the ugliest thing ever.

5. By which I mean, I wholeheartedly approve.

6. Of course, I forgot my camera, so I can`t take any photos to prove my point.

7. Mainz is like 2kmx2km.

8. Only the campus is like 200kmx200km.

9. This can only mean one thing: BIGGER ON THE INSIDE!!!!!!

10. Right in front of the Johannes Gutenberg Universität campus is a cemetery. I find it strangely fitting.

11. I haven`t been there yet, `cause they had no flats to offer there.

12. Not the sort of I was looking for anyway.

13. And none of them had an internet connection, so there.

14. I hate German keyboards. They have “y” where normal keyboards have “z” and vice versa. This is sick and perverted.

15. People are seriously so nice it almost creeps me out.

16. It also remains a mystery what they do on the weekends. Because, shops closed, restaurants and cafés closed, pubs closed, and only a small number of people can be found engaging in the wholesome activity of walking or cycling. WHERE IS THE REST? WORSHIPPING CTHULHU? Somehow, I wouldn`t be surprised. There is something about the small towns and the Great Old Ones, you know.

17. Lol, prime number!

18. Also, nobody cares if Polanski rots in prison. YAY.

Dear J-

Habits make you into a kind of unconscious zombie; whether you call it muscle memory or something more exotic, I’ve often found myself doing things with no rational explanation.  We fall into routines; I’ve been getting up early to go to work for ten years now and so the reflex to hit that snooze button is just something that happens automatically — I come to half-consciousness after half an hour of dozing and blindly stabbing out in the dark, with no particular memories of being disturbed.

We’ll have been in the house for six years in October; I could probably close my eyes at the front door and make it through without tripping or stumbing, as we haven’t significantly changed the furniture or layout since moving.  The house I grew up in didn’t feature a central hallway leading to various rooms; each room was connected to an adjoining one, so moving through there was an exercise in shutting doors behind you.  I got in the habit of grabbing the edge of the door just below the knob and pulling it behnd me; just before going off to college I noticed all the shiny patches on the edges where my hand had polished the varnish to a gloss.

When time is measured in milestones and memories rather than minutes and seconds, you start to see patterns repeating.  One of my favorite graphical mathematical constructs is the prime number spiral where you write the numbers in a square, outwardly-spiraling pattern (an Ulam Spiral):

101-100--99--98--97--96--95--94--93--92--91
 |                                       |
102  65--64--63--62--61--60--59--58--57  90
 |   |                               |   |
103  66  37--36--35--34--33--32--31  56  89
 |   |   |                       |   |   |
104  67  38  17--16--15--14--13  30  55  88
 |   |   |   |               |   |   |   |
105  68  39  18   5---4---3  12  29  54  87
 |   |   |   |    |       |  |   |   |   |
106  69  40  19   6   1---2  11  28  53  86
 |   |   |   |    |          |   |   |   |
107  70  41  20   7---8---9--10  27  52  85
 |   |   |   |                   |   |   |
108  71  42  21--22--23--24--25--26  51  84
 |   |   |                           |   |
109  72  43--44--45--46--47--48--49--50  83
 |   |                                   |
110  73--74--75--76--77--78--79--80--81--82
 |
111-112-113-114-115-116-117-118-119-120-121-...

If you then highlight the prime numbers (above, in bold red), you’ll begin to see that they tend to  lie out in diagonal patterns; prime number theory is itself fascinating, but one of the principles is that there is no mathematical formula to determine whether a given number is prime, and yet we find, graphically, that numbers seem to have habits despite the theories.  So embrace your habits; they serve you well.

Mike

I’ve been completing some of the problems posed in Project Euler recently. They’re quite good fun and most of the time its not just about getting the right answer but trying to get it done in the quickest time. A lot of the initial problems focus around prime numbers. I thought it would be nice to write a generic prime number generator which implements IEnumerable so you could just iterate through the list of prime numbers and do with them as you wish.

The class is based on the  Sieve of Eratosthenes, a quite common approach to generating a list of prime numbers. Most solutions I have seen so far required that you specify an upper limit for the prime number but I wanted to remove this requirement. I also wanted to constrain the memory footprint of the class so that it didn’t require a very large array to represent the entire sieve from 1 to n. The approach I took was to have a “sliding window”  which would represent the sieve. The sieve is initially setup with a window size of 1000 to represent the numbers 1 – 1001.  The usual approach of ruling out multiples etc. is handled in the GetEnumerator call which yields the next prime number on each iteration ( I do love the ease of writing an enumerator in C# )

The other condition the GetEnumerator function has to handle is the case when we have outgrown our sliding window. In this case we need to move the window along to the next n numbers and apply all the multiples of the prime numbers we have discovered so far. It works quite well and seems to be quite accurate but maybe I’ve missed an edge case or two. In terms of the memory footprint we have a constant sized array to represent our list of numbers, but we need to keep track of our list of prime numbers discovered so far in order to reapply their multiples when moving the window along. This is obviously going to continue to increase the more prime numbers we iterate through.

Here’s the code:

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;

namespace Problem10
{
    internal enum NumberType
    {
        Unchartered,
        Prime,
        Multiple
    }

    class EnumerableSieve : IEnumerable<int>
    {
        private List<NumberType> _numbers;
        private int _currentPosition;
        private readonly List<int> _primes = new List<int>( new[] {2} );
        private const int StepSize = 1000;

        public EnumerableSieve()
        {
            _numbers = new List<NumberType>(new NumberType[StepSize]);
            _numbers[0] = NumberType.Prime;
            _numbers[1] = NumberType.Prime;
        }

        public IEnumerator<int> GetEnumerator()
        {
            yield return _primes.Last();

            for(var lastPrime = _primes.Last(); ;lastPrime = _primes.Last() )
            {
                int multiplier = GetMultiplier(lastPrime);
                while (true)
                {
                    int multiple = GetMultiple(lastPrime, multiplier);
                    if (multiple < _numbers.Capacity)
                    {
                        _numbers[multiple] = NumberType.Multiple;
                        multiplier++;
                    }
                    else
                    {
                        var index = _numbers.FindIndex(number => number == NumberType.Unchartered);
                        if (index != -1)
                        {
                            int prime = AddPrime(index);
                            yield return prime;
                            break;
                        }

                        MoveWindow();
                        break;
                    }
                }
            }
        }

        private void MoveWindow()
        {
            _currentPosition++;
            _numbers = new List<NumberType>(new NumberType[StepSize]);
            ApplyPrimeComposites();
        }

        private int AddPrime(int index)
        {
            _numbers[index] = NumberType.Prime;
            int prime = GetPrime(index);
            _primes.Add( prime );
            return prime;
        }

        private int GetPrime(int index)
        {
            if (_currentPosition == 0) return index + 1;
            return (_currentPosition*StepSize) + index;
        }

        private int GetMultiple(int lastPrime, int multiplier)
        {
            var multiple = (multiplier * lastPrime) - (_currentPosition * StepSize);
            if (_currentPosition == 0) multiple -= 1;
            return multiple;
        }

        private void ApplyPrimeComposites()
        {
            foreach (var prime in _primes)
            {
                var i = (int)Math.Ceiling((double)(_currentPosition*StepSize)/prime);
                while (true)
                {
                    var multiple = (i*prime) - (_currentPosition*StepSize);
                    if (multiple < _numbers.Capacity)
                    {
                        _numbers[multiple] = NumberType.Multiple;
                        i++;
                    }
                    else
                    {
                        break;
                    }
                }
            }
        }

        private int GetMultiplier( int lastPrime )
        {
            if (_currentPosition != 0)
            {
                return (int)Math.Ceiling((double)(_currentPosition * StepSize) / lastPrime);
            }

            return 2;
        }

        IEnumerator IEnumerable.GetEnumerator()
        {
            return GetEnumerator();
        }

    }
}

Recently I noticed that Jenny’s phone number 8675309 (867-5309) had some interesting number theoretic properties.

So, this evening I decided to punch my own phone number into Wolfram|Alpha to see what I’d find. Amazingly, both my 7-digit phone number and my full 10-digit phone number are prime numbers! That’s so excellent. (Unfortunately, putting a 1 in front of it makes it composite.)

According to the prime number theorem, the chance that a number is prime is approximately . Thus, the chance that my 7-digit phone number is prime is approximatly 7% and the chance that my 10-digit number is prime is about 4.4%. So the chance that they are both prime is approximately 0.3% (assuming they are independent, which I guess they aren’t since knowing that one is prime implies that the other one is at least odd, so maybe the chance of both being prime is more like 0.6%…?)

Actually, that got me to thinking, what is the longest prime number with the property that if we drop the left-most digits one-at-a-time, we always have primes? I looked at this list of primes and found 96823. We see that 3, 23, 823, 6823, and 96823 are all primes. My guess is that we can find such a number of arbitrary length.

Speaking of primes, this whole thread began when I posted a Nova ScienceNow video about the twin prime conjecture. One of my colleagues pointed out that the filmmaker erred by implying that 1 is a prime number. See the screenshot below with the primes colored red. Oops.