Maths Ed Ideas: Prime Numbers

Showing posts with label Prime Numbers. Show all posts

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)?

You may find PeakBagger.com and PeakList.org useful starting points to collect your data. WolframAlpha may also help.

Monday, 8 January 2018

On Frustrating Primes

7775353777 is a prime number. It has ten digits and it looks like, on first inspection, that the second five digits reflect the first. But a number cannot be prime and have such a property — and on closer inspection we can see that there is, indeed, no such reflection. Such primes can be considered 'frustrating' because of the absence of this visually eye-catching, aesthetically pleasing pattern they appear, on the initial face of it, to promise.

A prime, therefore, is said to be frustrating if the first half of its digits occur again in the second half, but without a) the order repeating (i.e. abc‖abc, because this is divisible by 1001 [1] and thus not prime), or b) the order reflecting (i.e. abc‖cba, because every palindrome with an even number of digits is divisible by 11 and thus not prime).

As such, a frustrating prime has an even number of digits, and contains at least six digits of which at least two are distinct [2]. Of the 68,906 six-digit primes [3], 265 are frustrating (collated in the image above). The possible arrangements of six-digits with the first three reoccurring:

aaa‖···

⇒ aaa‖aaa : Not prime ∵ divisible by 11

aab‖···

⇒ aab‖aab : Not prime ∵ divisible by 1001

⇒ aab‖aba : Possibly a frustrating prime (n = 10)

⇒ aab‖baa : Not prime ∵ divisible by 11

aba‖···

⇒ aba‖aab : Possibly a frustrating prime (n = 7)

⇒ aba‖aba : Not prime ∵ divisible by 1001

⇒ aba‖baa : Possibly a frustrating prime (n = 10)

abb‖···

⇒ abb‖abb : Not prime ∵ divisible by 1001

⇒ abb‖bab : Possibly a frustrating prime (n = 9)

⇒ abb‖bba : Not prime ∵ divisible by 11

abc‖···

⇒ abc‖abc : Not prime ∵ divisible by 1001

⇒ abc‖acb : Possibly a frustrating prime (n = 66)

⇒ abc‖bac : Possibly a frustrating prime (n = 58)

⇒ abc‖bca : Possibly a frustrating prime (n = 68)

⇒ abc‖cab : Possibly a frustrating prime (n = 37)

⇒ abc‖cba : Not prime ∵ divisible by 11

The Frustrating Primes are recorded as sequence A297994 in the On-line Encyclopedia of Integer Sequences (OEIS), from which you can find this table of Frustrating Primes a_n for n = 1 to 10,000.

The learning capital to be accrued from exploring the idea of frustrating primes, is especially evident in terms of the development of students' mathematical maturity. The above sketch of the underlying premise of frustrating primes informs a 'thinking through' of how as teachers we could use the idea with (alongside) students, how we could approach the problem with students, structurally and pedagogically, and what questions can/should be asked of our students, should they not yet be able to ask them for themselves.

Considering the idea through this problem will encourage depth and maturation, and give us as teachers the opportunity, as I wrote about in a previous post, to 'model for [students] what it is to be mathematically mature, to behave in a mathematically mature manner when we are befuddled by the beautiful sixes and sevens of uncertainty.' Students will not only deepen their sense of number and their appreciation of prime numbers, at the same time as being able to apply divisibility tests, consider prime factors, etc., but they may also get a kick out of doing something a little more mathematically novel than they are typically used to.

Notes & (Select) Links:

[1] \(\left( {1000 \times \overline {abc} } \right) + \left( {1 \times \overline {abc} } \right)\)

[2] For a two-digit number to be a frustrating prime the first half of the digits would have to occur in the second, and this would thus be a number of the form a‖a, which is clearly a composite number divisible by 11. Similarly, for a four-digit number to be a frustrating prime the first half of the digits would have to occur in the second, meaning that the number would thus be of the form ab‖ab or ab‖ba, which are both composite as the former is divisible by 101 (100×ab+1×ab) and the latter is an even palindrome and thus divisible by 11.

[3] For a full list (via Wolfram Alpha)