On 31 August someone posting as kbsriram opened a project on hackaday.io called A Tiny Sieve. Thirty ATtiny412s, one modulus per chip, standing in a 3U Eurocard stripboard backplane and sharing a single open-drain VOTE line, with a Raspberry Pi Pico clocking them and checking whatever survives. The design he is copying is D. H. Lehmer's 1982 electronic sieve, with the ATtinys standing in for Lehmer's ROM-and-counter pairs. The project description is unusually direct about why. From the 1920s to the late 1990s, the fastest way to answer a certain class of number-theory question was not a general-purpose computer but a special-purpose machine called a number sieve; the Lehmers — father, son, and the son's wife, all three mathematicians — built them first from bicycle chains and gears, then delay lines, then shift registers, then custom VLSI. "Nothing in chains and gears does arithmetic, so how does it work? And why did they last so long into the age of computers? I built one to find out."

It really does not do arithmetic. Each chip holds one row of yes and no: for its modulus p, which residues could still be a square. On every clock it moves its finger one cell along, and if that cell says no it pulls the shared line low; at the end of the row it starts again. The line stays high only when all thirty are silent. The machine never produces an answer. It produces silence.

Start with where the construct sits. The cut that made the general-purpose computer fell across the question of whether a machine has to be shaped like its problem. The stored program is revolutionary precisely because it lets a machine be shaped like nothing, and therefore like anything, and the whole history of computing is narrated forward from that point. What fell on the other side of the cut was the machine shaped like the problem itself: thirty gears are thirty moduli, one wheel per modulus. It has no slot to go in. It is not a computer, not a calculator, not an analogue computer, and it does no arithmetic at all. The awkward part is that it kept winning. The DLS-127 began operating in 1965 and ran twenty-four hours a day for ten years; the whole machine cost about two thousand dollars, of which about a hundred and fifty went on navy-surplus wire delay lines; its input was a paper tape twenty-five feet long carrying 28,077 characters, punched by a CDC 6400 — the general-purpose machine employed here as a tape punch for the special one. The line runs on to a 1995 doctoral thesis at the University of Manitoba titled, flatly, A Very Fast Electronic Number Sieve. Seventy years alongside the computer, faster than it at the one thing it did, and outside the story.

But the thing worth looking at is not the speed. It is that the machine refuses to conclude. A sieve excludes only what is provably impossible: if a number lands on a non-residue mod p it cannot be a square, and that cut is clean and never wrong. Beyond that it says nothing. What survives is not an answer, it is what nobody has managed to veto, and it has to be handed to something slower to be checked. kbsriram states the rule in a line: every sieve has something slower behind it, or the sieve would be no use. The numbers are eloquent. Running three moduli against Lehmer's sixteen-digit number 5,283,065,753,709,209, the sieve scanned 5,676,028 candidates in a hundred and thirteen seconds and let about seven hundred and forty thousand through; the Pico spent twenty-three minutes testing them one at a time before returning 59,957 and 88,114,244,437. Across the whole run the sieve was idle ninety-two per cent of the time, waiting on its checker. The 1928 machine of nineteen bicycle chains did sixty candidates a second and a human checked the survivors. The 1932 machine of thirty gears, which stopped when a beam of light found its way through aligned holes onto a photocell, did five thousand a second and a human still checked the survivors. The machine got eighty times faster and the checking end did not move. The remainder always outruns the naming.

Here is the sharp part: a veto leaves no trace. A machine that can only say no cannot tell you which time it said no wrongly. kbsriram puts it plainly — with thirty chips on one open-drain line, a single card holding that line low because of one solder bridge will reject every candidate that should have passed, and the run will complete perfectly normally and report that no solution exists. This is not a failure that raises an error. It is a failure that impersonates a result. So he built an entire apparatus whose only job is to certify that the vetoes were honest: CRC-8 on the I2C commands; a one-byte response code on every command, so a card can refuse a corrupted instruction instead of silently obeying it; a STATUS command returning the card's ring position, whether it is armed, its modulus, its firmware version and the reason for its last reset, so he can see whether a card browned out mid-run; four electrical checks at flash time — shorts to ground, pin bridges, opens, isolation — with no production firmware written to a card that fails one; and a full thirty-card self-test before every run. The boards cost eighty-one dollars for thirty with shipping and tariffs, about $2.70 a modulus, and their headers are two hundred and forty solder joints, any one of which is enough to make a run fail in silence.

There is a counterpart to this from 1932. That September the elder Lehmer described his son's photoelectric gear machine in Scripta Mathematica: the light passing through the aligned holes fell on the photocell for a ten-thousandth of a second, and that impulse had to be amplified seven hundred and twenty-nine million times to cut the motor. The amplifier was accordingly so highly strung that it was moved behind a brick wall and packed into what he called a coffin of celotex, and still threw a fit every quarter of an hour. Nothing in its symptoms explained it. In the end he attached a loudspeaker and listened to it, and found a short-wave radio enthusiast transmitting in the neighbourhood. That kind of error is at least recoverable: the machine stops, you wind the gears back looking for the beam, you find no beam, and you know it lied. The danger in the modern machine runs the other way. A false positive halts it and can be inspected. A false negative does nothing at all, looks exactly like there being no solution, and can never be found. In SAE terms: severing destroys the evidence of what it severed, which is why severing and cultivation are not two symmetrical operations. Cultivation leaves behind the thing it grew. Severing leaves nothing behind, including no record of its own mistakes. The only defence is a second ledger kept outside the machine, by hand, recording what should have been rejected this time, and checked afterwards against what was. Most of kbsriram's five build logs are that ledger.

As for how long a remainder can sit where it is, this lineage has already answered. The first genuinely successful automatic congruence machine was not a Lehmer's. It was Eugène-Olivier Carissan's, completed in 1919 by the Paris horologists Chateau Frères, hand-cranked, and able to prove 1,321,442,641 prime in fifteen minutes. Both Carissan brothers were dead by 1925 and the machine slid into obscurity; D. H. Lehmer, who spent his life building sieves, first heard of them in 1989; the machine entered the Musée des Arts et Métiers in 1994, and the paper announcing its reappearance was called "Discovery of a Lost Factoring Machine." Seventy years — inside a field of a few dozen people worldwide, with everyone who would have cared already looking.

Seeing this now matters more than seeing it later, for specific reasons. As I write, the project stands at sixty-five views, one follower, three likes and no comments. Created 31 August, last updated a day ago, five logs. It is mid-build: the thirty cards have only just passed their first whole-backplane self-test, and the full thirty-modulus run has not happened yet — the reason the project exists, the thing it is meant to find out, is still in the future tense. It is entered in Hackaday's Retro Computing Contest, and the slot marked retrocomputing is already open and waiting for it. But it does not fit. Retrocomputing runs on collective memory, on machines enough people touched to be nostalgic about. Nobody has a childhood memory of a number sieve; perhaps a few dozen people ever used one. That is the gap the naming leaves. His own stated motive is a question rather than a conclusion: I built one to find out. Go while it is still being debugged, because every card carries an LED, and what it blinks is each veto that card casts. He has made no into something you can watch.

Han Qin (秦汉) · @hqinjarsy · SAE Judgment & Aesthetics