June 1, 2026
Prime drama, composite chaos
Only 17% of all 64-bit Integers are products of two 32-bit integers
Math fans are stunned that most giant numbers never show up in simple multiplication
TLDR: A programmer showed that only about 17% of all 64-bit numbers can be made by multiplying two 32-bit numbers, which matters because simple computer scrambling methods can miss huge chunks of possible outputs. The comments swung between amazement, nitpicking, and jokes about "fixing math," turning a niche number fact into comedy.
A delightfully nerdy math post somehow turned into a full-on comment-section variety show after readers learned a weird fact: if you multiply two ordinary 32-bit numbers and keep the full result, you can only make about 17% of all possible 64-bit numbers. Translation for the rest of us: even when you use two very large everyday computer numbers, most bigger numbers are still unreachable. And yes, people absolutely had feelings about that.
The biggest split in the crowd was between the mind blown camp and the well, actually squad. One side was marveling that a random 64-bit number will "usually fail" to be expressible this way. The other side immediately barged in to say, hold on, multiplication ignores order, so of course the number of possible outcomes collapses fast. Classic internet scene: one group staring into the void, the other group bringing a spreadsheet. Meanwhile, another commenter won the thread with mock-activist energy, joking, "Together, we can change math for the better," as if 64-bit integers just need better representation.
There were also attempts to make the whole thing feel less alien. One commenter compared it to two-digit numbers: surprisingly few can be made by multiplying two one-digit numbers. Another tossed out a spicy connection to Benford’s Law, because no math thread is complete without someone opening a whole new rabbit hole. The actual article is about why this matters for hash functions—ways computers scramble data—but the comments turned it into the real spectacle: half awe, half nitpicking, all deliciously nerdy chaos.
Key Points
- •The article studies the exact fraction of 64-bit unsigned integers that can be written as the product of two 32-bit integers.
- •It motivates the question through hash-function design, using a simple multiplication-based hash example and referencing the author's clhash work.
- •It cites Erdös for the asymptotic result that the proportion of reachable 2n-bit values from multiplying two n-bit values tends to zero as n grows.
- •For the smaller 16-bit-to-32-bit case, the article says slightly more than one out of five 32-bit numbers can be produced by multiplying two 16-bit integers.
- •Using mathematics and code attributed to Webster and colleagues, the article reports that exactly 3,215,709,724,700,470,902 unsigned 64-bit integers—about 17% of all 64-bit values—are products of two 32-bit integers.