(This page was first written in 2017 - each year I make additions. Perhaps it may be of interest to people who teach elementary number theory, or go on long train journeys.)
This year's date is 2025 = 45^2, and is therefore easy to factor:
2025 = 3^4 * 5^2. (3^4 means 3 to the power 4).
But there is more to say about this number:
2025 = (20 + 25)^2 = (1 + 2 + ... + 9)^2 = 1^3 + 2^3 + ... + 9^3.
The equality of the last two expressions is an example of the Theorem of Nicomachus (c.60 - c.120):
1^3 + 2^3 + ... + N^3 = (1 + 2 + ... + N)^2 = ( N*(N+1)/2 )^2.
There is pretty webpage about this:
https://en.wikipedia.org/wiki/Squared_triangular_number
Here are the factors of last year's date:
2024 = 2^3 * 11 * 23.
What can we say about 2024? This is a number of the type: n*(n+1)*(n+2)/6. In our case n = 22. These are called tetrahedral (or triangular pyramidal) numbers. 2024 is the number of balls in a triangular pyramid of side 22. At the top, there is 1 ball. Below that, on the second level, there are 3 balls. On the third level, 6 balls. On the n-th level, there are n*(n+1)/2 balls. On the 22nd level there are 22*23/2 = 253 balls. In total there are 2024 balls. There are many other interesting things about this number. To find them, put 2024 into the search box of the Online Encyclopedia of Integer Sequences:
A tetrahedral date is a rare event! The last one (with n=21) was 1771, the next one (n=23) will be in 2300.
Here are two recent dates which are palindromes: 1991, 2002. But 2023 was a different kind of palindrome:
2023 = 7 * 17 * 17.
The year before was:
2022 = 2 * 3 * 337.
The number 2022 occurs in this interesting sequence:
H4: 1, 2, 3, 4, 5, 6, 7, 510, 1014, 2022, ....
A Harstad number (or Niven number) is one divisible by the sum of its digits. The numbers 1, 2, ... , 10 are obviously Harstad numbers. The number 2022 is a more interesting example, being divisible by 6, as we stated above. Such numbers are common. But a run of 4 consecutive Harstad numbers is rare. 2022 is the start of such a run - you can easily check that the numbers 2022, 2023, 2024 and 2025 are divisible by 6, 7, 8 and 9 respectively.
The sequence H4 consists of just those numbers which are the start of a run of 4 Harstad numbers You can find this sequence listed in the Online Encyclopedia of Integer Sequences (see link above), and learn what numbers come next ...
The year before was 2021 = 43 * 47 (one way to discover this is mentioned below).
The factors are consecutive primes. Such a year is again a rare event. The last one was 1763 = 41 * 43. The next will be 2491 = 47 * 53.
The year before that was 2020 = 2(2) * 5 * 101.
The year before that, 2019, was three times a prime (3*673), the first such year of the millenium.
The year before that, 2018, was twice a prime (2*1009), also the first such year of the millenium.
The year before that, 2017, was a prime, the 3rd of the millenium, but the first of the form 4k+1 (see below).
The year before that, 2016, was smooth (lots of small factors):
2016 = 2^5 * 3^2 * 7.
This number is divisible by all the numbers d = 2, 3, 4, 6, 7, 8, 9. It follows that the date 2016 + d has the factor d for all these values. This explains why 2022 is in the sequence H4!
Three years before that all had three prime factors:
2015 = 5 * 13 * 31,
2014 = 2 * 19 * 53,
2013 = 3 * 11 * 67.
How do you find these facts in your head? Let us consider showing that 2017 is prime. Here is one of many ways to test if 13 divides 2017. Add 13, divide by 10, subtract 13, divide by 10 again. The result is 19, not divisible by 13 - so neither is 2017 (since each step we took does not change whether 13 is a factor). Of course, if we already knew that 13 was a factor of 2015, it cannot be a factor of 2017 also. Another easy way in this case is to note that 2017 = 15 (mod 1001 = 7 * 11 * 13), so none of 7, 11 and 13 can be factors. To show that 2017 is prime, we have to verify that none of primes 2, 3, 5, ... , 43 is a factor - the next prime 47 exceeds the square root of 2017 (which is below 45, see below).
Here is a general way to test if a prime p(>5) divides a given odd number:
Add or subtract p or an odd multiple of p, divide out the powers of 2, also 3 or 5 if present, and continue (until we have a number so small that we can easily see if p is a factor).
Example. To test if 31 divides 2017, just add 31 and recognise the result as a power of 2.
Note.
i. It is always possible to get a final zero by adding or subtracting p or 3p - this may or may not be the easiest way to proceed.
ii. It may also be worthwhile to subtract a large even multiple of p, e.g. 10p, 100p, etc.
As we said above, this year is a rare event, a date which is a perfect square: 2025 = 45^2. The last time was 1936 = 44^2. As a result, some years in this millenium can be factored at once by Fermat's method:
2024 = 45^2 - 1^2 = 44 * 46 = 2^3 * 11 * 23.
2021 = 45^2 - 2^2 = 43 * 47.
2016 = 45^2 - 3^2 = 42 * 48 = 2^5 * 3^2 * 7.
2009 = 45^2 - 4^2 = 41 * 49 = 7^2 * 41.
(The first three of these results were already given above).
Now primes, apart from 2, are odd and so of one of the forms 4k + 1 or 4k +3. The year 2017 was the third prime of the millenium but the first of the form 4k + 1, the first two, 2003 and 2011, being of the other form 4k + 3. So this is our first chance to demonstrate Fermat's Two Square Theorem - that primes of form 4k + 1 are the sum of two squares (and in just one way):
2017 = 44^2 + 9^2.
Primes of form 4k + 3 are not a sum of two squares - that is quite simple, since no integer of form 4k + 3 is the sum of two squares (for an odd sum, one square must be odd and one even, but then the sum is of form 4k + 1).
There were 168 primes in the first millenium, 135 in the second, and so far 3 in the third, so 2017 is the 306th prime.
(Last modified 13th January 2025)