[unwind]

Cool, and even cooler that the idea is due to Knuth.

I found this typo entertaining:

Technically it could be replaced by any other large prime number. The most important thing is that it must have few factors, and be large enough to distribute information into the higher value bits when the integer overflows.

Surely any prime number has "few factors", i.e. none other than itself and 1? I guess the "prime" in the first sentence is a typo, since the second sentence reads as if the first one didn't say "prime".

[FabHK]

My understanding was that for it to work, it must have few factors, and in particular no small ones, so easiest just to chose a large prime number.

[quinnftw]

I noticed that too, I think he may be referring to the fact that generating super large actual prime numbers is hard, so people use probabilistic algorithms to generate probably almost primes

[Someone]

This one isn't superlarge, by a wide shot. √2,654,435,761 = 51521.2166102, so even an extremely naive trial division loop needs just over 50k trial divisions.

I haven't benchmarked it, but I think that should easily run in under a second on any phone sold in the last 10 years. If you do it a bit smarter and skip even divisors, I think it could run in under a second on 30 year old hardware (25k iterations, 200-ish cycles per loop (division was very expensive, back then) takes a 5MHz CPU)

[mark-r]

The problem wasn't in proving this particular number was prime, it was finding a large prime number to begin with. I don't know what the state of the art was back in the day, but 32 bits would certainly have been too large for the Sieve of Eratosthenes.

[Someone]

No, it wasn’t. ln(2654435761) is about 22, so if you limit yourself to odd numbers, about one in 11 attempts near that number will give you a prime.

At a generous one minute to prove primality of numbers in that range using trial division, and assuming you forget to bail out early when you find the first divisor, you still very likely will find a prime within an hour.

Even in the 70s, when computer time cost real money, that shouldn’t deter you.

(I just tried this on a Mac Mini, in Swift; finding 1000 primes took about a third of a second; 3 seconds if I forget to bail out early)

Edit: as further evidence that this isn't that large a prime: we knew 2^31-1 to be prime by trial division in 1772 (Euler somehow found time to do that)

[dfox]

For context: factor(2654435761) on original series Ti-89 takes almost exactly two seconds.

[quinnftw]

This particular one isn't large, but in general a pseudo-prime will produce similar effects.