happy number

Happy Numbers

Happy numbers are defined by the rule that you start with any positive integer, square each of the digits then add them together.  Now do the same with the new number.  Happy numbers will eventually spiral down to a number of 1.  Numbers that don’t eventually reach 1 are called unhappy numbers.

As an example, say we start with the number 23.  Next we do 2²+3² = 13.  Now, 1²+3² = 10.  Now 1²+o² = 1.  23 is therefore a happy number.

There are many things to investigate.  What are the happy numbers less than 100?  Is there a rule which dictates which numbers are happy?  Are there consecutive happy numbers?  How about prime happy numbers?  Can you find the infinite cycle of sadness?

Nrich has a discussion on some of the maths behind happy numbers.  You can use an online tool to test if numbers are happy or sad.

perfectnumber Perfect Numbers

Perfect numbers are numbers whose proper factors (factors excluding the number itself) add to the number.  This is easier to see with an example.

6 is a perfect number because its proper factors are 1,2,3 and 1+2+3 = 6

8 is not a perfect number because its proper factors are 1,2,4 and 1+2+4 = 7

Perfect numbers have been known about for about 2000 years – however they are exceptionally rare.  The first 4 perfect numbers are 6, 28, 496, 8128.  These were all known to the Greeks.  The next perfect number wasn’t discovered until around 1500 years later – and not surprisingly as it’s 33,550,336.

The next perfect numbers are:

8,589,869,056 (discovered by Italian mathematician Cataldi in 1588)

137,438,691,328 (also discovered by Cataldi)

2,305,843,008,139,952,128 (discovered by Euler in 1772).

and they keep getting bigger.  The next number to be discovered has 37 digits are was discovered over 100 years later.  Today, even with vast computational power, only a total of 48 perfect numbers are known.  The largest has 34,850,340 digits.

There are a number of outstanding questions about perfect numbers.  Are there an infinite number of perfect numbers?  Is there any odd perfect number?

Euclid in around 300BC proved that that 2p−1(2p−1) is an even perfect number whenever 2p−1 is prime.  Euler (a rival with Euclid for one of the greatest mathematicians of all time), working on the same problem about 2000 years later went further and proved that this formula will provide every even perfect number.

This links perfect numbers with the search for Mersenne Primes – which are primes in the form 2p−1.  These are themselves very rare, but every new Mersenne Prime will also yield a new perfect number.

The first Mersenne Primes are

(22−1) = 3

(23−1) = 7

(25−1) = 31

(27−1) = 127

Therefore the first even perfect numbers are:

21(22−1) = 6

22(23−1) = 28

24(25−1) = 496

26(27−1) = 8128

friendlynumber

Friendly Numbers

Friendly numbers are numbers which share a relationship with other numbers.  They require the use of σ(a) which is called the divisor function and means the addition of all the factors of a.  For example σ(7) = 1 + 7 = 8 and σ(10) = 1 +2 +5 + 10 = 18.

Friendly numbers therefore satisfy:

σ(a)/a = σ(b)/b

As an example (from Wikipedia)

σ(6) / 6 = (1+2+3+6) / 6 = 2,

σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2

σ(496)/496 = (1+2+4+8+16+31+62+124+248+496)/496 = 2

Therefore 28 and 6 are friendly numbers because they share a common relationship.  In fact all perfect numbers share the same common relationship of 2.  This is because of the definition of perfect numbers above!

Numbers who share the same common relationship are said to be in the same club.  For example, 30,140, 2480, 6200 and 40640 are all in the same club – because they all share the same common relationship 12/5.

(eg. σ(30) /30  = (1+2+3+5+6+10+15+30) / 30 = 12/5 )

Are some clubs of numbers infinitely big?  Which clubs share common integer relationships?  There are still a number of unsolved problems for friendly numbers.

solitarynumber

Solitary Numbers

Solitary numbers are numbers which don’t share a common relationship with any other numbers.  All primes, and prime powers are solitary.

Additionally all number that satisfy the following relationship:

HCF of σ(a) and  a = 1.

are solitary.  All this equation means is that the highest common factor (HCF) of σ(a) and a is 1.  For example lets choose the number 9.

σ(9)= 1+3+9 = 13.  The HCF of 9 and 13 = 1.  So 9 is solitary.

However there are some numbers which are not prime, prime powers or satisfy HCF (σ(a) and  a) = 1, but which are still solitary.   These numbers are much harder to find!  For example it is believed that the following numbers are solitary:

10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99

But no-one has been able to prove it so far.  Maybe you can!

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.