Embed Size (px)
Citation preview
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Assorted Topics: Prime Numbers and OtherProblems
Sarah Brandsen, Erin Conley
Duke University
20 November, 2019
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Prime numbers!
Today we’ll spend time thinking about an interestinggroup of numbers: prime numbers
In particular, we will consider...I Interesting patterns that arise in prime numbersI Ways to look for prime numbers (Mersenne primenumbers, spirals)
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Prime numbers: interesting patterns!
Lots of patterns in the prime numbers!
Here’s a pattern we will prove: For any prime number p, thesquare p2 is 1 more than a multiple of 24, i.e., p2 = 24n + 1 forinteger n.
I Note: this doesn’t work for “sub-primes” 2 or 3
Example: 52 = 25 = 24+ 1What about 7? 17? (Hint: 172 = 289)
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
p2 = 24n + 1
For p = 7 and p = 17:I 72 = 49 = 24× 2 + 1I 172 = 289 = 24× 12 + 1
How do we prove this is true for generic p?
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Before we prove p2 = 24n + 1
In order to prove p2 = 24n + 1, we want to learn moreabout p and multiples of 6:
Things we care about: prime numbers, multiples of 6
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Primes on a number line
Multiples of 6 are sandwiched between primes!Although this is not always the case (e.g., 24, where23 is prime but 25 is not)
This is just a fancy way of saying that prime numbersdon’t have 2 or 3 as a factor
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Now onto the proof for p2 = 24n + 1
So we have discovered that primes (not subprimes) are either 1above or below a multiple of 6, i.e., p = 6k ± 1 for integer k
k will either be odd or even, i.e., k = 2m pr k = 2m + 1
So we have four categories of primes:I p = 6k + 1 for even k : p = 12m + 1I p = 6k + 1 for odd k : p = 12m + 7I p = 6k − 1 for even k : p = 12m − 1I p = 6k − 1 for odd k : p = 12m + 5
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Proving p2 = 24n + 1
Now we square the four categories and prove thateach category obeys p2 = 24k + 1:
(12m + 1)2 = 144m2 + 24m + 1 = 24(6m2 +m) + 1
What about p2 = (12m + 7)2?
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Proving p2 = 24n + 1
I (12m+ 7)2 = 144m2 + 168m+ 49 = 24(6m2 + 7m+ 2)+ 1I (12m − 1)2 = 144m2 − 24m + 1 = 24(6m2 −m) + 1I (12m+ 5)2 = 144m2 + 120m+ 25 = 24(6m2 + 5m+ 1)+ 1
Since we covered every possible prime (and made the problemeasier by splitting up the primes into categories), we haveproved that p2 = 24n + 1 X
But there is an “easier” way...
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Proving p2 = 24n + 1 another way...
Is p2 − 1 a multiple of 24?I p2 − 1 = (p + 1)(p − 1)
Consider p − 1, p, p + 1 on a number line:I p has no factorsI p must be odd; either p − 1 or p + 1 is a multiple of 4 →the product (p − 1)(p + 1) will be a multiple of 8
I Either p − 1 or p + 1 will be a multiple of 3, which means(p − 1)(p + 1) will be a multiple of 3
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Proving p2 = 24n + 1 another way...Since (p − 1)(p + 1) is both a multiple of 3 and a multiple of 8,then it must also be a multiple of 24!
Note: we didn’t use the fact that p is prime (we used p is oddand p is not divisible by 3) → we have really proven that for allnumbers that don’t have 2 or 3 as a factor, their square is amultiple of 24 plus 1
Another note: this proof is easier in a mathematical sense(easier in hindsight)
For more information, see Squaring Primes - Numberphile
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
How do you find primes?
Interested in looking for primes - what are some ways of doingthat?
The largest known prime number was found by the GreatInternet Mersenne Prime Search (GIMPS) team, a network ofcomputing people who look for large prime numbers
I Largest prime: 257,885,161 − 1I 57, 885, 161 is the 48th Mersenne prime to ever be found
For more information, see New Largest Known Prime Number -Numberphile
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Mersenne primes
Mersenne prime: 2p − 1 for some prime p
The three largest known primes (2p − 1):I p = 57885161I p = 43112609I p = 42643801
How many digits does the largest prime have?
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Largest prime number fun facts!
257,885,161 − 1 has 17,425,170 digits!
From this prime, we can also deduce a perfect number(i.e., a positive integer equal to the sum of its divisors,e.g., 6 = 3+ 2+ 1): 257885160 × (257885161 − 1) (whichhas about 34 million digits)!
Fun fact: you could get a lot of money ($150,000 ormore) for finding big primes! Learn more here.
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Another way to find primesNow let’s consider square spirals:
What happens when you pick out the prime numbers?
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Primes on a square number spiral
Prime numbers seem to lineup along diagonals, i.e.,some diagonal lines containmore primes than others
Could we look for primes onthose lines? Is it random?
Right: 200× 200 square
Image courtesy of Ulam spiral - Wikipedia
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Is it random?
Image courtesy of Prime Spirals - Numberphile
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Finding equations for the diagonal linesUsing the spirals might beanother way to find primesbecause you can makeequations for the diagonal
I Example: main diagonalgoing up and to the right(see image) can be writtenas y = 4x2 − 2x − 1
I When x = 2, y = 13;x = 3, y = 31 X
I What about x = 4?Image courtesy of Prime Spirals - Numberphile
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Finding lines with high prime density
Spiral diagonals seem to indicate some quadratic equationsproduce more prime numbers than others...?
I This is a conjecture that hasn’t been proven; it’s based onanecdotal findings
I Example: some lines has 7 times as many primes; thelargest discussed in Prime Spirals - Numberphile was 12times as many primes (compared to average)
I Example “golden line”: x2 + x + 3399714628553118047
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
An aside on the square number spiralUlam spiral: named after StanisławUlam, a Polish mathematician andnuclear physicist
Said to have doodled the spiral in1963 during a presentation about“a long and very boring paper”
He and other collaborators used afirst-generation computer to extendthe spiral out to 100, 000 points
Image courtesy of Stanislaw Ulam - Wikipedia
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
An aside on the square number spiralFeatured on the cover ofScientific American in1964!
If you’re interested incoding and visualizingthe Ulam spiral, thencheck out Fun with theUlam spiral
Image courtesy of Fun with the Ulam spiral
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Another spiral you can makeInstead of a square spiral, nowconsider a spiral where all squarenumbers form a line
Form the spiral by drawing onethat connects the square numbers,and evenly space the remainingnumbers along the spiral
What happens with you circle theprimes? Image courtesy of The Sacks Number Spiral
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Picking out the primes
Image courtesy of The Sacks Number Spiral
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
More patterns!
Gap for the square numbers
Develop formulas for the curvesto find primes? If we can betterunderstand the patterns, thenwe could solve importantconjectures in math (e.g., twinprime conjecture)
Image courtesy of Ulam spiral - Wikipedia
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
An aside on the Sacks spiral
Named after Robert Sacks(software engineer); hedevised his spiral in 1994
He consideredArchimedean spirals whereone counterclockwiserotation produces the nextsquare number
Image courtesy of Archimedean spiral - Wikipedia
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
An aside on the Sacks spiralSacks focuses on productcurves: lines thatoriginate from (or near)the spiral center; theselines traverse the spiralarms at different angles
Could these curves be usedto predict large primenumbers? It’s currently anopen question
Image courtesy of The Sacks Number Spiral
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Competition-Style Math- Example 1
Problem- “Several sets of prime numbers, such as{7, 83, 421, 659} use each of the nine nonzero digits exactlyonce. What is the smallest possible sum such a set of primescould have?”
artofproblemsolving.com
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Competition-Style Math- Example 1Problem- “Several sets of prime numbers, such as{7, 83, 421, 659} use each of the nine nonzero digits exactlyonce. What is the smallest possible sum such a set of primescould have?”
Solution- Neither of the digits 4, 6, and 8 can be a units digitof a prime. Thus,
sum ≥ 40+ 60+ 80+ 1+ 2+ 3+ 5+ 7+ 9= 207
We can group these to form the set of primes {41, 67, 89, 2, 3, 5}
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Competition-Style Math- Example 2
Problem- Solve the equation
cosn x − sinn x = 1
where n is a given positive integer.
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Competition-Style Math- Example 2
Problem- Solve the equation cosn x − sinn x = 1 where n is agiven positive integer.
Solution- We know that cos2 x + sin2 x = 1, such that therecannot any solutions where both cos x and sin x are nonzero.
We then look for equations where cos x = 0 or sin x = 0, soI for even n- x = mπ for integer mI for odd n- x = mπ for integer m or x = m 3π
2
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Birthday Problem
Problem- Suppose you are in a room with 23 people chosen atrandom. What is the probability that at least two people share abirthday?
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Birthday ProblemSolution- denote P(share) as the probability that at least twopeople share a birthday, and P(no share) as the probability thatno two people share a birthday. Then:
P(share) = 1− P(no share)
and
P(no share) =364365× 363
365× ...× 365− (23− 1)365
= 0.4927
So the probability is about 50.7%!
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Birthday Problem GraphLet’s see how this probability scales with the number of peoplein the room:
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Two Children Paradox
Consider the following questions:I Mr. Jones has two children. The oldest child is a girl.What is the probability that both children are girls?
I Mr. Smith has two children. At least one child is a boy.What is the probability that both children are boys?
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Two Children Paradox
Let’s start with the first question: Mr. Jones has two children.The oldest child is a girl. What is the probability that bothchildren are girls?
Solution- pretty straightforward solution here! The youngerchild is a girl with probability 0.5. So the probability that bothchildren are girls is 0.5.
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Two Children Paradox
Mr. Smith has two children. At least one child is a boy. What isthe probability that both children are boys?
Solution- the solution is actually a mystery due to unclearphrasing! We can interpret this two ways:
I We meet one of Mr. Smith’s children at random, and learnthis child is a boy (not knowing anything about theremaining child). This gives probability 0.5
I Mr. Smith tells us that, out of his two children, at leastone is a boy
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Two Children Paradox
Let’s consider the second case where Mr. Smith tells us that,out of his two children, at least one is a boy. Then theprobability that both are boys is 1
3 as shown:
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Two Children Paradox Variant
Let’s consider one final question about Mr. Smith!
Problem- “Mr. Smith is the father of two. We meet himwalking along the street with a young boy whom he proudlyintroduces as his son. What is the probability that Mr. Smith’sother child is also a boy?”
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Hall’s Marriage Theorem
Problem- Suppose you have x students looking for jobs from xcompanies. Each student will take any job they can get,whereas not every company will hire every student. Giveninformation on which students each company will hire, how canwe determine if there is a “perfect matching” solution?
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Hall’s Marriage Theorem
Problem- example graph for 4 students and 4 companies. Isthere a match here?
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Hall’s Marriage Theorem
Solution- no match in this case, as Blizzard and Google bothonly want Corki (but Corki can only work for one company!)
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Hall’s Marriage Theorem
Theorem- “For a set of n companies, denote m to mean thenumber of students that at least one of these companies want.If m ≥ n for every set of companies, then a matching ispossible. Otherwise, the matching fails.”
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Hall’s Marriage Theorem– Application
Putnam Problem- “Suppose 2m teams play in a round-robintournament. Over a period of 2m-1 days, every team plays everyother team exactly once. There are no ties.
Show that for each day we can select a winning team, withoutselecting the same team twice.“
Assorted Topics:Prime Numbers
and OtherProblems
Sarah Brandsen,Erin Conley
1.) Primenumbers!
2.) p2 = 24n + 1
3.) Mersenneprimes
4.) Ulam spiral
4.) Ulam spiral
6.) Competition-styleProblems
7.) Two ChildParadox
8.) Hall’sMarriageTheorem
www.egmon.com.br
Thank you!
Thank you for coming to Math Circles!
