Tuesday, 14 August 2018

On Descartes Numbers

A number is said to be Perfect if it is equal to the sum of its proper positive factors, i.e. the sum of its factors excluding the number itself.  Summing all of the factors of a number \(n\) — i.e. including the number \(n\) itself — is known as the sum-of-divisors function:

\[\sigma \left( n \right) = \sum\limits_{d\left| n \right.} d \]
A number \(n\) is thus said to be Perfect when \(\sigma \left( n \right)\) = 2n.  For example, the factors of 28 — 1, 2, 4, 7, 14 and 28 — sum to 28 + 28 = 56, thus \(\sigma \left( {28} \right)\) = 2 \(\times \) 28 and so 28 is a Perfect number.  In contrast, and by way of illustration, the factors of 42 — 1, 2, 3, 6, 7, 14, 21 and 42 — sum to 96, or 2.2857... \(\times \) 42, thus \(\sigma \left( {42} \right) \ne\) 2 \(\times \) 42 and so 42 is not a Perfect number.  In fact, 42 is said to be an Abundant number because the sum of its proper positive divisors is greater than the number itself, i.e. \(\sigma \left( n \right) > 2n\).  When the sum of a number's proper positive divisors is less than the number itself, i.e. \(\sigma \left( n \right) < 2n\), the number is said to be Deficient.  For example, the factors of 50 — 1, 2, 5, 10, 25 and 50 — sum to 93, or 1.86 \(\times \) 50, and thus \(\sigma \left( {50} \right)\) < 2 \(\times \) 42.  As such:

\[\begin{array}{l}\sigma \left( n \right) < 2n \Rightarrow n\;{\rm{is}}\;{\rm{deficient}}\\\sigma \left( n \right) = 2n \Rightarrow n\;{\rm{is}}\;{\rm{perfect}}\\\sigma \left( n \right) > 2n \Rightarrow n\;{\rm{is}}\;{\rm{abundant}}\end{array}\]
Perfect numbers have intrigued us since Euclid, who observed via Proposition 36 in Book IX of his Elements that a number of the form (2\(^{p - 1}\))(2\(^p\) − 1) is a perfect number whenever 2\(^p\) − 1 is prime (i.e. what subsequently became known as a Mersenne prime) [1].  The fact that every even perfect number is of this type, i.e. that every even perfect number can be written in the form (2\(^{p - 1}\))(2\(^p\) − 1) whenever 2\(^p\) − 1 is prime, was proposed by René Descartes in his famed correspondence with Marin Mersenne (specifically in his letter to Mersenne of 15 November 1638) [2], and later proven in 1849 by Leonhard Euler [3].  Descartes was, indeed, 'among the first to consider the existence of odd perfect numbers' (Greathouse and Weisstein, 2012).

As of writing, of the fifty perfect numbers that have been found, all are even, and it is not known if there are infinitely many or, indeed, whether any odd perfect numbers exist — although Pascal Ochem and Michaël Rao showed (2012) that no number up to 10\(^{1500}\) is an odd perfect number.

One number has been found, however, that would have been an odd perfect number if only one of its factors was prime rather than a 'spoof prime' (i.e. a composite number wrongly assumed to be prime).  This number — 198,585,576,189 — was found in 1638 by Descartes, documented in his 15 November letter to Mersenne, and is known as a Descartes Number, or, as an 'odd spoof perfect number' (note that the spoof prime in question is 22021, i.e. 19\(^2\) \(\times \) 61).

\[198,585,576,189 = {3^2} \times {7^2} \times {11^2} \times {13^2} \times 22021\]
Descartes showed that if 22021 was prime, the proper factors of 198,585,576,189 [4] would sum to 198,585,576,189 — making it the only odd perfect number ever found.  As it is, because 22021 is not prime, or because 22021 is a spoof prime, the proper factors of 198,585,576,189 do not sum to 198,585,576,189 (they actually sum to 227,441,894,589).  This makes 198,585,576,189 an odd spoof perfect number, which in itself is the only such number ever found!

To elaborate: Descartes showed that an equivalence to the sum-of-divisors function using the prime factorisation of a number is the following [5]:

\[\sigma \left( n \right) = \prod\limits_{{p^a}\left\|{\;n} \right.} {\frac{{{p^{a + 1}} - 1}}{{p - 1}}} {\rm{ }}\]
where \(p\) is a distinct prime factor.  This means that a number's factors can be summed using only the number's prime factors.  Taking 28 again as an example, we have seen that it is a perfect number because

\[\sigma \left( 28 \right) = \sum\limits_{d\left| 28 \right.} d  = 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2n\]
We can also observe this result using the prime factors of 28, i.e. 2\(^2 \times \) 7:

\[\prod\limits_{{p^a}\left\|{\;n} \right.} {\frac{{{p^{a + 1}} - 1}}{{p - 1}}}  = \frac{{{2^{2 + 1}} - 1}}{{2 - 1}} \times \frac{{{7^{1 + 1}} - 1}}{{7 - 1}} = \frac{7}{1} \times \frac{{48}}{6} = 56 = 2n\]
And so, similarly, the sum of the divisors of Descartes' number:

\[\begin{array}{c}\begin{align}\sigma \left( 198,585,576,189 \right) &= \prod\limits_{{p^a}\left\| {\;n} \right.} {\frac{{{p^{a + 1}} - 1}}{{p - 1}}} \\ \\&= \frac{{{3^{2 + 1}} - 1}}{{3 - 1}} \times \frac{{{7^{2 + 1}} - 1}}{{7 - 1}} \times \frac{{{{11}^{2 + 1}} - 1}}{{11 - 1}} \times \frac{{{{13}^{2 + 1}} - 1}}{{13 - 1}} \times \frac{{{{22021}^{1 + 1}} - 1}}{{22021 - 1}}\\  \\&= 13 \times 57 \times 133 \times 183 \times 22022\\ &= 13 \times \left( {3 \times 19} \right) \times \left( {7 \times 19} \right) \times \left( {3 \times 61} \right) \times \left( {2 \times 7 \times 11 \times 11 \times 13} \right)\\ &=2 \times {3^2} \times {7^2} \times {11^2} \times {13^2} \times \left( {{{19}^2} \times 61} \right)\\ &= 2 \times {3^2} \times {7^2} \times {11^2} \times {13^2} \times 22021\\ &= 2 \times 198,585,576,189\end{align}\end{array}\]
Hence, if 22021 were prime rather than 19\(^2\) \(\times \) 61, Descartes' number 198,585,576,189 would  be an odd perfect number because \(\sigma \left( n \right)\) = 2n.

Notes, References & Links:

Wednesday, 11 July 2018

Calendar of Mathsy Moments

The calendar of mathsy 'moments' (downloads provided below) is a calendar marking at least one mathsy 'moment' (or event) of interest for each day from 1 August 2018 to 31 December 2019.  It is primarily intended for schools and teachers to use in gentle cultivation of a respectful yet intellectually playful approach to mathematics in their students, in support moreover of the development of students' deeper and perhaps wider appreciation and framing of mathematics in the historical, cultural and intellectual sense.

The calendar is shared to help teachers establish and cultivate an ethos of mathematics as progress in the classroom, to help teachers in other words help their students develop and sustain happy and fulfilling relationships with their mathematics, encouraging the desire to explore further, deeper, and more independently — fuelling students' curiosity and their capacity for such.  It is hoped that in sharing these brief moments of mathematics with students, in unabashedly celebrating the joy that is to be found in mathematics, teachers will fulfil their roles as enthusiasts, advocates and celebrators of mathematics, both as an intellectual discipline and cultural artefact, and students will in turn stand a greater chance that the subject will inspire — rather than beguile — them.

The calendar is provided below for access from this page.  Downloads in a range of formats are also provided for easy import.  Each 'moment' is outlined in more detail within the description / notes section of each calendar entry, which may serve as a prompt for the teacher with respect to how s/he may use / share it, if at all.  Where applicable, links to sources, and other sources of information — including online articles, historical artefacts, videos, podcasts, etc. — are provided within each description as a start for further exploration, as desired.

The moments are categorised within the calendar as follows:
  • Moments / Events occurring at various times throughout the year (note that a 'Day' relates to the day of the year, e.g. 14 March 2019 is the 73rd day of the year, and so a curio about 73 may be given.  A 'Date' relates to the date as a concatenated number, in dd/mm/yyyy format unless otherise indicated, e.g. 14 March 2019, the 14th day of the 3rd month in the 2019th year, 14/3/2019, which concatenates to 1432019).  These moments include:
    • Prime Number Dates, Mathematicians' Birthdays, Historical Dates to note in Mathematics, Circular Prime Number Dates, Emirp Dates, Frustrating Prime Number Dates, Days for Curious Calculations, Days for Number Facts, Palindromic Number Dates, Palindromic Prime Number Days, Pi Dates, et al.
    • The 'Day for a Number Fact' days aims to feed our natural love of numbers (or perhaps revitalise it), exposing students to — or maybe tantalising them with — a wide range of more exotic sounding mathematics from number theory that they may not ordinarily come across in their day to day curriculum.  For example, in sharing such moments students will know of: good primes, home primes, long primes, Sophie Germain primes, Ramanujan primes, Fibonacci primes, near-repdigit primes, quadruprimes, prime triplets, prime sums, primes that remain prime when added to their reverse, circular primes, emirps, permutable primes, near primorial primes, palindromic primes, dihedral primes, twin primes, sexy primes, numbers commonly assumed incorrectly to be prime, semiprimes, sphenic numbers, other prime products, unprimeable numbers, happy numbers, sad numbers, distinctly happy numbers, perfect numbers, superperfect numbers, pseudoperfect numbers, weird numbers, abundant numbers, deficient numbers, almost integers, duck-duck-goose numbers, Mersenne numbers, lucky numbers, amicable numbers, polite numbers, sociable numbers, automorphic numbers, narcissistic numbers, odious numbers, pernicious numbers, evil numbers, practical numbers, arithmetic numbers, polygonal numbers, undulating numbers, telephone numbers, antisigma numbers, highly composite numbers (or anti-primes), refactorable numbers, taxicab numbers, tetranacci numbers, vampire numbers, untouchable numbers, Bell numbers, Euclid numbers, Franel numbers, Frobenius numbers, mountain numbers, star numbers, Smith / Joke numbers, Friedman numbers, the divisor (factor) function, the partition function, Jacobi's formula, Aronson's sequence, Fibonacci factorials, Euler's 6n+1 theorem, repdigits, primorials, subfactorials, double factorials, superfactorials, hyperfactorials, trees, Golomb's sequence, numbers in different bases, etc.
  • Moments / Events occurring once in the year (including some national or globally recognised awareness days with particular reference to mathematics).  These moments include:
    • A Perfect Day, An Anti-Perfectt Day, Ada Lovelace Day, e day (Europe and US), Fibonacci Day (US), Half Year Day, Halloween (Vampire Numbers), Mathematics and Statistics Awareness month, McNugget Numbers Day, National Numeracy Day, Phi Day, Phi Day (US), Pi Approximation Day, Pi Day, Root-2 Date, Spreadsheet Day, The Golden Ratio Moment, The Oddest Prime Day, et al.
    • Other Moments / Events not directly related to mathematics (that schools / teachers may nonetheless wish to mark, either because of the mathematics latent within them, or for other more school-centred reasons around values, community and relationships).  These moments include: 
      • Ask A Stupid Question Day, Autumnal Equinox, Black History Month, Book Lovers' Day, European Day of Languages, International Joke Day, International Women's Day, Make Up Your Own Holiday Day, National Coding week, National Poetry Day, National Read a Book Day, National School Nurse Day, National Teaching Assistant's Day, Origami Day, Random Act of Kindness Day, Roald Dahl Day, Space Day, Summer Solstice, Vernal Equinox, Winter Solstice, World Book Day, World Kindness Day, World Music Day, World Teachers' Day.

    Mathsy Moments calendar
    (Click here to go back to the top)
    There are and will be, of course, many other 'moments' that could and maybe should be included in such a calendar.  I make no claim that the calendar offers anything comprehensive.  If there are other 'moments' that you think other teachers may enjoy sharing with students, please let me know in the comments and I will update in subsequent versions.  Equally, whilst every effort has been taken to eliminate errors, if users notice any error in the content of the calendar, I would be very grateful if you could again let me know in the comments.

    GCSE Maths 2019 Countdown calendar
    (Click here to go back to the top)
    This calendar gives a daily countdown to GCSE mathematics exams, both in terms of days left and school days left.  It is intended as a support for teachers to help students (and their families) manage their own study time.

    Further sources
    Sources of information (or explanation), where applicable, have been provided in link form in the description of each moment / event.  If you wish to play further, explore the following sites:

      Tuesday, 1 May 2018

      On Mountain Numbers

      Cairn Toul, a beautifully wild and remote mountain in the Cairngorms massif of the eastern Scottish highlands, is notable not only for being the fourth highest mountain in the UK, but also for being the only mountain in the UK whose elevation in metres — 1291m above sea level — is also a 'mountain'.  Großglockner, by way of a further example, the highest mountain in Austria, is a mountain whose elevation in feet — 12461ft above sea level — is also a 'mountain'.

      To elaborate: A number is a 'mountain' if its decimal digits start with 1, i.e. 'base camp', ascend continuously to a unique summit, i.e. to one largest digit, then descend continuously back to 1, i.e. back to base camp.  Such mountain numbers are recorded as sequence A134941 (itself a mountain) in the On-line Encyclopedia of Integer Sequences (OEIS), from which you can also find this table of all possible 21,846 mountains (including 1 itself).

      Mountain primes, as the name suggests, are prime mountain numbers, prime numbers in other words whose decimal digits start with 1, ascend continuously to the summit of a single largest digit, then descend continuously back to 1.  (The area chart image above shows a mountain range of the first 17 such numbers.)  Mountain primes are recorded in the OEIS as sequence A134951 (yes, also a mountain prime), from which you can find this table of all 2620 such primes.

      When climbing a mountain, it is typical to descend back to base camp, i.e. back to where you started your ascent, but it is not always the case.  Mountain numbers that ascend from one location (as an elevation) but descend to another — e.g. 3,598,432 — are considered to be generalised mountain numbers, recorded as sequence A134853 in the OEIS (there are 173,247 such mountains).

      Suggested explorations, diversions & links:
      • Explain why there must be a finite number of mountain numbers.
      • Construct a mountain range diagram for mountains of your choice.
      • What is the Everest of mountain numbers?
        • How will you define 'Everest'; how will you define 'elevation'?  (In the image above, the 'height' of mountain 1291, for example, is greater than the 'height' of mountain 1571.)  
      • Explore the distribution of mountain number digits, i.e. how many mountains have 1 digit, 2 digits, 3 digits, etc. (OEIS sequence A135417).
      • If we regard the number of digits in a mountain number as the horizontal distance travelled when climbing such a mountain:
        • What mountain(s) has (have) the shallowest ascent?
        • What mountain has the steepest ascent and descent?
        • What about generalised mountains?
      • Many people find the most beautiful mountains to be those most pyramidal in shape, such as for example the Matterhorn in the Alps and Machapuchare in the Himalaya.
        • Define and find your most beautiful mountain numbers: 
          • Palindromic mountain numbers perhaps (OEIS sequence A173070),
          • Or palindromic mountain prime numbers (the largest is 123467898764321).
        • Find all possible — i.e. 45 — Giza numbers.
          • Giza numbers are so-called because they represent the pyramids of Giza in the sense that their first digits increase in consecutive order to a largest central digit, and their last digits decrease in the same consecutive order as they increased.  
          • The largest Giza number is 12,345,678,987,654,321 (OEIS sequence A134810).
      • Of two mountain numbers chosen at random, what is the probability that their numerical heights — i.e. the magnitude of the number — will be in alignment with the elevation of actual mountains?  
        • For example, mountain number 1291 is 'higher' than mountain number 1571, but as actual elevations, 1291m < 1591m, so these two mountains are not in 'alignment'. 
      • Find as many actual mountains whose heights are mountain numbers, or mountain primes.
        • What is the highest mountain in the world whose height is also a mountain (use metres and/or feet for the height)?