Okay, I’ve got this thing about prime numbers. Some people get all woozy for holiday music as December marches along, but the turning of the year puts me into numeric mode. I’ve done year‑end posts about the special properties of the integer 2016 and integers made up of 3s and 7s. (Sheldon Cooper’s favorite, 73, is just part of an interesting crowd.)

I looked up “2025” in the On-line Encyclopedia of Integer Sequences (the OED of numbers). That number is involved in 1028 different series or families. Sequence A016754, the Central Octagonal Numbers, has some fun visuals. Draw a dot. Then draw eight dots symmetrically around it. You have nine dots. Nine is O₂, the second Central Octagonal Number (an octagon enclosing a center, such a surprise). It’s ‘second‘ after O₁=1, for that first dot. Now draw another octagon of dots around the core you started, but with two dots on each side. Those 16 dots plus the 9 inside make 25, so O₃=25. An octagon with three dots on each side has 24 dots so O₄ is 1+8+16+24=49 (see the figure). And so on. If you do the arithmetic, you’ll find that O₂₂, the 22nd Central Octagonal Number, is 2025. Its visual has 22 rings (including the central dot), 168 dots in its outermost ring, for 2025 dots in all.

In case you’re wondering, there is a non-centered series of octagonal numbers that grow out of a dot placed at a vertex of a starter octagon. 2025 isn’t in that series. See the hexagon equivalent in my 2015 post.

Sadly, 2025 isn’t a prime year. Prime‑number years, 2003 and 2011 for example, can be evenly divided by no integer other themselves (and one, of course). 2017 was a prime year, but we won’t see another until 2027. Leap year numbers are divisible by 4 so they can’t ever be prime. That property disqualified 2020 and 2024. It’ll do the same for 2028 and 2032.

Two primes that are as close together as possible, separated only by a single (necessarily even) number, are called twins. There were no twin‑prime years in the 700s, the 900s or the 1500s. The thirteen prime years in the twenty‑first century include three sets of twins, 2027‑2029, 2081‑2083 and 2087‑2089.

If a number’s not prime, then it must be divisible by at least two factors other than itself and one. 2018 and 2019, for example, each have just two factors (2×1009 and 3×673, respectively). Numbers could have more factors, naturally — 2010 is 2×3×5×67 and 2030 is 2×5×7×29, four factors each.

A single factor could be used multiple times — 2024 is 2×2×2×11×23, also written as 2³×11×23, for a total of 5 factors. We’re just entering a 6‑factor year (see below) but a formidable factor‑champion is on the horizon. Computer geeks may be particularly fond of the year 2048, known in the trade as 2k (not to be confused with Y2K). The number 2048 has eleven factors, more than any year number of last or this millennium. 2048 is 2¹¹, the result of eleven 2s multiplied together. Change just one of those 2s to a 3 and you have 3072 which is a long time from now.

So anyhow, I was poking at 2025, just seeing what was in there. The 5 at the tail‑end is a dead give‑away non‑prime‑wise because the only prime that ends in a 5 is … 5. Another useful trick – add up the digits. If the sum is divisible by 3, so is the number. If the sum is divisible by 9 so is the number. Easy to figure 2+0+2+5=9, so two easy ways to know that 2025‘s not prime.

By the time I got done breaking the number down into all six of its factors, look what a pretty pattern appeared:

Finally, 2025 appears 8 times in this post’s text. Happy New Year.

~~ Rich Olcott