THE FOLD / RESPAWN / THE RESURRECT / THE WHEEL FACTORISATION
THE WHEEL FACTORISATION
skip what cannot possibly be prime
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A wheel of the first few primes skips every number they divide. Roll it and only the survivors need testing. The number of spokes per revolution turns out to be exactly Euler’s totient of the circumference — the sieve and the number-theoretic function are the same count.
LIT verified live. A {2} wheel has circumference 2 and 1 spoke; {2,3} gives 6 and 2; {2,3,5} gives 30 and 8; {2,3,5,7} gives 210 and 48. Every spoke count equals φ of the circumference, 4/4. And the returns shrink: each new prime removes 16.7%, then 6.7%, then 3.8%, while the table grows from 2 entries to 210.
LIT verified live. A {2} wheel has circumference 2 and 1 spoke; {2,3} gives 6 and 2; {2,3,5} gives 30 and 8; {2,3,5,7} gives 210 and 48. Every spoke count equals φ of the circumference, 4/4. And the returns shrink: each new prime removes 16.7%, then 6.7%, then 3.8%, while the table grows from 2 entries to 210.
2 HOW IT WAS WEAVED · AI + HUMAN
Wheel factorisation is the standard optimisation on top of trial division and the Sieve of Eratosthenes — the familiar “check 2, then only odd numbers” is the {2} wheel, and “6k ± 1” is the {2,3} wheel written out.
AVAN (AI) checked the totient identity rather than assuming it, because it is the reason the wheel has a closed form at all: the spokes are exactly the residues coprime to the circumference, and counting those is what φ does. What that buys is a prediction — the next wheel, {2,3,5,7,11}, has circumference 2310 and φ = 480, so 20.8% survive for an eleven-fold table. The diminishing return is not an observation about these four; it is what φ(n)/n does as you multiply in more primes.
AVAN (AI) checked the totient identity rather than assuming it, because it is the reason the wheel has a closed form at all: the spokes are exactly the residues coprime to the circumference, and counting those is what φ does. What that buys is a prediction — the next wheel, {2,3,5,7,11}, has circumference 2310 and φ = 480, so 20.8% survive for an eleven-fold table. The diminishing return is not an observation about these four; it is what φ(n)/n does as you multiply in more primes.
3 ONE DIMENSION
Four wheels, and the totient beside each.
4 TWO DIMENSIONS · INTERACTIVE
Add a prime to the wheel and watch the spokes thin.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a wheel with most of its spokes removed.
AVAN’s addition (the inverse-companion): the forward reading is “a bigger wheel skips more work.” The inverse is that the survivors are not primes and the wheel never claims they are. A {2,3,5,7} wheel passes 121, 143 and 169 straight through — every product of primes above 7 survives every wheel that can be built. Read backwards, the wheel removes only the cheapest composites, the ones a single division would have caught anyway, and leaves the entire hard part of the problem exactly where it was.
LIT a {2} wheel has circumference 2 and 1 spoke, {2,3} gives 6 and 2, {2,3,5} gives 30 and 8, and {2,3,5,7} gives 210 and 48 - every spoke count equalling phi of the circumference, 4 of 4; and the returns shrink, each new prime removing 16.7%, then 6.7%, then 3.8%, while the table grows from 2 entries to 210
FIG Wheel factorisation is the standard optimisation on top of trial division and the Sieve of Eratosthenes - the familiar 'check 2, then only odd numbers' is the {2} wheel, and '6k plus or minus 1' is the {2,3} wheel written out. AVAN checked the totient identity rather than assuming it, because it is the reason the wheel has a closed form at all: the spokes are exactly the residues coprime to the circumference, and counting those is what phi does. What that buys is a prediction - the next wheel, {2,3,5,7,11}, has circumference 2310 and phi = 480, so 20.8% survive for an eleven-fold table.
FIG Wheel factorisation is the standard optimisation on top of trial division and the Sieve of Eratosthenes - the familiar 'check 2, then only odd numbers' is the {2} wheel, and '6k plus or minus 1' is the {2,3} wheel written out. AVAN checked the totient identity rather than assuming it, because it is the reason the wheel has a closed form at all: the spokes are exactly the residues coprime to the circumference, and counting those is what phi does. What that buys is a prediction - the next wheel, {2,3,5,7,11}, has circumference 2310 and phi = 480, so 20.8% survive for an eleven-fold table.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN