Happy New Year 2019! (And Mathematical Facts About 2019)

Time by moments steals away,
First the hour, and then the day;
Small the daily loss appears,
Yet it soon amounts to years.

— John Newton, The Works of the Rev. John Newton, 1827

As the new year approaches, so is the second anniversary of Learn Fun Facts. But before that, let’s examine some numerical curiosities concerning the number 2019.

Every year, I try to find a “countdown” pattern for a particular year. For the year 2019, I’m happy to say that I have found a few countdown patterns, albeit not as impressive as last year’s:

happy new year 2019 math.png

The first two ones are straightforward while the third one is a bit more complex. For the last one, it has a couple of factorials. I won’t bore you with its formal definition. It just basically means:

5! = 1 × 2 × 3 × 4 × 5
6! = 1 × 2 × 3 × 4 × 5 × 6

In my opinion, the last one is not as elegant as the first two since it contains parentheses. I have found more examples during my search for countdown patterns but I chose to not include them as they are too contrived.

Palindromic Magic Square for 2019

18 12 28 16 06 27

The above magic squares only used the digits 2, 0, 1, and 9 from “2019”. They are palindromic in nature so one is the reverse of the other. Their magic sums are the same, which is 132. They also share additional patterns and you can learn more about them in this post.

Some Large Primes

  • 2 × 102019 – 87 is prime (A273907).

Pandigital New Year

In number theory, pandigital numbers are numbers that contain all the digits in a given base. To not complicate matters, I would just focus on base 10 numbers. For base 10, a pandigital number should have the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. To illustrate, 1234567890 and 7430125689 are pandigital numbers.

For the year 2019, I found the following sum of exponents with pandigital digits 0-9 and no redundant digits:

pandigital exponents 2019.png

Note that 7⁰ is equal to 1.

Another minor pandigital coincidence involving 2019 is this:

PANDIGITAL NEW YEAR FOR 2019.png

Happy-Go-Lucky Number

A happy number is found using the following procedure: Pick any positive integer (e.g. 97).Calculate the sum of its digits (9² + 7² = 130). Continue until you end up with one of these ten numbers: 0, 1, 4, 16, 20, 37, 42, 58, 89, or 145. The only way to get 0 is if you start with the number 0. If you get 1 (where it would stay, since 1² = 1), the number is said to be a happy number. In our example, the resulting number is 1:

9² + 7² = 130 → 1² + 3² + 0² = 10 → 1² + 0² = 1.

Therefore, 97 is a happy number.

If the chosen number resulted in any of the eight numbers: 4, 16, 20, 37, 42, 58, 89, or 145, it will loop in a cycle among these numbers. Any number which ends up in any of these numbers after performing the procedure is called a sad number or unhappy number.

Here are the first few happy numbers (A007770):

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100 …

A lucky number is determined using a sieving process. The sieve involves eliminating numbers based on their position in the remaining set. To perform the sieving process, first, write down a list of consecutive whole numbers starting with 1:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 …

Eliminate every second number (i.e. every even number):

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 …

Eliminate every third number in the remaining numbers:

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 …

1, 3, 7, 9, 13, 15, 19, 21, 25 …

Eliminate every seventh number in the remaining numbers, which starts with 19:

1, 3, 7, 9, 13, 15, 19, 21, 25 …

1, 3, 7, 9, 13, 15, 21, 25 …

After the sieving process, all the numbers that remained are called lucky numbers. The first few lucky numbers are (A000959)

1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111 …

A number that is both a happy number and a lucky number is called happy-go-lucky number and 2019 just happens to be one!

So, have a Happy-Go-Lucky New Year!

A Little Curiosity

2019-math-new-year.png

It was quite a surprise when I found this one.

Sum of Perfect Powers

Perfect power is any whole number that can be expressed as a power of another whole number. For example, 125 is a perfect power because it’s equal to 5³.

2019 is a sum of the first 22 perfect powers:

2019 = 1 + 4 + 8 + 9 + 16 + 25 + 27 + 32 + 36 + 49 + 64 + 81 + 100 + 121 + 125 + 128 + 144 + 169 + 196 + 216 + 225 + 243

Note: The perfect powers sequence can be found in A001597.

Sum of Three Squares

Legendre’s three-square theorem asserts that a positive integer n is a sum of three squares if and only if it is not of the form of n=4^a(8b+7) for integers a and b. 2019 can’t be represented in that form so, this means that it can be written as the sum of three squares. As it turns out, there are nine cases for this:

1² + 13² + 43² = 2019
5² + 25² + 37² = 2019
7² + 11² + 43² = 2019
7² + 17² + 41² = 2019
11² + 23² + 37² = 2019
13² + 25² + 35² = 2019
17² + 19² + 37² = 2019
23² + 23² + 31² = 2019

Happy Cube Root New Year

I think that this is rather contrived, but I decided to share this here anyway:

cube root new year 2019.png

Miscellaneous 2019 Facts

  • 2019² = 1155² + 1656²
  • Using 1, 2, 3, and 4: 2019 = 4 × 21² + 12 × 21 + 3.
  • Sum of Four Squares: 2019 = 17² + 23² + 24² + 25².
  • Sum of Five Squares: 2019 = 15² + 17² + 20² + 23² + 24².
  • Sum of Six Squares: 2019 = 15² + 16² + 17² + 18² + 21² + 22².
  • Difference of Two Squares: 2019 has a couple of cases:

338² – 335² = 2019
1010² – 1009² = 2019

  • Some Exponential Patterns:

2019 = 15² + 28² + 1² + 28² + 15²
2019 = 1³ + 7³ + 11³ + 7³ + 1³

  • 2019 = 225 × 3² – (2 × 3).
Advertisements

Posted by

My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. You can find me on Twitter` and Facebook. My email is edmarklaw@learnfunfacts.com

71 thoughts on “Happy New Year 2019! (And Mathematical Facts About 2019)

  1. Thank you visiting my blog and giving my latest post a like. I’m glad you enjoyed Santiago Oaks. Happy New Year, and congratulations on the 2nd anniversary of LFF! I enjoy browsing your eclectic site.

    Like

What's On Your Mind?

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s