Early morning, but an August morning so it’s hot already. I’m thanking the Acme Building’s air conditioning as I unlock the door to my 12th‑floor office. There’s Vinnie, stretched out on the couch with a book. “What’re you reading there, Vinnie?”
“Morning, Sy. It’s one of Ken Follett‘s books, Pillars of The Earth, about a couple of families battling over a cathedral. One family wants to build it, the other hates whatever the first family wants, and this goes on for generations. Meanwhile, I’m learning more than I want to know about how they made felt in Medieval times.”
“Follett does like to dive down rabbit holes, and he’s proud to show off what he digs up. I just finished another of his books, Code to Zero. It’s a thriller loaded with arcane tidbits. The action’s wrapped around some Soviet plot in the late 1950s to disrupt America’s Explorer satellite program. We get close-ups of rocketship anatomy; we find out why unsymmetrical dimethylhydrazine is a bad bet as a rocket fuel; things like that.”
“Sounds interesting.”
“Yeah, I think you’d like it. Lots of action while the hero’s chasing around trying to figure out what happened to him and why. Spoiler alert — things get frantic toward the end.”
“Thrillers always do, that’s why they’re thrillers.”
“Mm‑hm. But I’m a numbers guy, so what really caught my eye was Follett’s collection of obscure numerical lore. One of the book’s characters, Elspeth, has an ongoing math‑nerd game with another character. Each player names a special number. The other has to explain why that number’s special.”
“Can’t be too hard. Practically every number’s special somehow.”
“True, but Follett found some doozies. For instance, Elspeth plays number 29. It’s special because it’s the root of a whole train of prime numbers: 29, 31, 37, 47,… all united by a common formula. The nth member of the series (counting 29 itself as n=0) is given by 29+2n2. The series breaks at n=29 because 29+2*292=1711, which clearly is divisible by 29.”
“29 days in February, sometimes.”
“That’s not number‑theory special. Another Elspeth gambit involves the abc=a1+b2+c3 formula where each letter is a base‑10 digit on one side and a number raised to a power on the other. She plays 135, special because 11+32+53=1+9+125=135. The other nerd matches her pattern with 11+72+53=1+49+125=175. That got me wondering if any other numbers fit the template. I worked it up on Old Reliable—”
“Of course you would.”
“You know me so well. Turns out that both 518 and 598 qualify, but no others. Elspeth plays with cubes, too. There’s a conversation in the book about 136 and 244 being partners because 13+33+63=244 but 23+43+43=136. Number 8000 is 203 but it’s also 113+123+133+143.”
“How do they even find these things?”
“My guess is heroic mathematicians with time on their hands in a no‑machinery no‑screens environment. Think, these guys were doing arithmetic with an abacus. Four centuries ago Fermat conjectured that numbers of the form 22n+1 are prime. He tried cases n=0 through 4 and they all worked. A century later, Euler showed that 225+1=4 294 967 297 isn’t prime — it’s evenly divisible by 641. Imagine even generating that number on an abacus. Then imagine looking for a zero remainder each time you abacus‑divide by potentially 65000 successive primes. Fortunately Euler struck gold on the 115th test.”
“Rechecked it a couple of times, I bet.”
“Indeed. Those cubes — number theoreticians love sums of powers. There’s one example in Code to Zero that boggles my mind. Up near the book’s frantic finale, Elspeth just happens to know that 168303 is the sum of the 1000 cubes from 11343 through 21333. Why would some hero hit on that combination, and why would they bother to check if the sum is a cube?”
“Dunno, I’m not a numbers theory guy.”
~ Rich Olcott
- Thanks to Lloyd Boyer for pointing me to Code to Zero and the wonders therein.
Hard Science Ain't Hard 