If you’ve followed this blog for more than a year (in which case, thanks), you’ve probably noticed that I like numbers. As a kid I got a thrill when I figured out why (111 111 111)² is the amazingly symmetrical 12 345 678 987 654 321. Like many other numberphiles I’m fascinated by special categories like primes (no divisor other than unity and the number itself) and figurate numbers (counting the dots that make a geometrical pattern like concentric hexagons).

Every number has charms of its own (2025 is a delicious example), but 2026 doesn’t seem like it’s particularly special. 2026 is 2×1013, so it’s not prime nor is it the square or power of some smaller number. It’s not palindromic like 1991 and 2002 are. In binary it’s 111 1110 1010₂ — not very interesting. It can be written as 20₁₀₁₃ , which is just silly. But 2026 does turn out to be special, in a special way — it’s a Beprisque number and they’re rare.

The On-line Encyclopedia of Integer Sequences®, a.k.a. OEIS, does for whole numbers what Bartlett’s Familiar Quotations does for words: it’s a venerable compendium of usage in context, with citations. An OEIS search for “2026” found 270 sequences that include the number. Most are technical or abstruse to a fault. Entry A118454, for example, identifies 2026 as the algebraic degree of the onset of the logistic map’s 11th bifurcation. Don’t ask.

Among my search hits was A216840, “palindromic numbers of 3 digits in two bases.” It’s cool that 2026₁₀ is bcb₁₃ and also a4a₁₄. Unfortunately, my math software’s number‑base converter has only 36 available digits (0‑9, a‑z) so it can’t display numbers in base‑37 or worse.

My favorite entry in the set is A163492: Beprisque numbers are integers such that their two immediate neighbors on the number line are a perfect square and a prime. 10 is Beprisque because it’s nestled between 9=3² and 11 (a prime). So is 2026, sitting between 2025=45² and 2027 (prime).

“Beprisque” — looks kind of French, doesn’t it? Maybe named after one of those 18th century scientist‑aristocrats that Robespierre killed off? Nope, it’s simply a mash‑up of “between prime and square.”

This being a New Year post, I couldn’t resist mentioning the New Year monkey. Every year the financial press services trot the poor thing out, give it a handful of darts with instructions to fling them it at the Wall Street Journal‘s stock listing or some such. The monkey’s acumen is assessed by how much its dart‑selected equities have grown (or not) over the past year. Usually the monkey’s score rates near or even better than that of the “professional” stock pickers, which is supposed to tell us something.

Suppose we were to direct the monkey to toss its darts at the number line. How well would it do at hitting Beprisque numbers? How dense are the primes and squares and what are the odds that one from one category is exactly two seats away from one from the other?

It depends on how much of the line the monkey can aim at. The range between 1 and 100 has 10 squares (10% coverage), 25 primes (25% coverage) and 9 Beprisques (1, 2, 3, 8, 10, 24, 48, 80, 82). Nine out of a hundred is 9% coverage; looks good. But widen the range to 1000 and the coverages drop — 31 squares (3%), 168 primes (17%) and 17 Beprisques — 10 times the range but fewer than twice the hits. Extend the number line to 1 000 000 and the percentages get even worse: 0.1% squares, 7.9% primes and only 0.022 6% Beprisques. The Beprisque coverage drops to 0.007 2% if the monkey’s target range shifts to the span between one and two million.

Why so few? On the average primes are further apart for bigger numbers because the prime count grows about 15% more slowly than the range does. Meanwhile, the separation between successive squares grows linearly as the squared number increases:
  (n+1)² – n² = n²+2n+1 – n² = 2n+1
The two trends conspire to thin out those cozy sequences. Give the poor monkey an infinite range and its odds on a square‑Beprisque‑prime triplet evaporate like a certain team’s championship hopes.

~ Rich Olcott