Expand description
Infinite integers generated by the program’s generator H (dt = 1/H) — the trip counts of
supertasks.
A loop that steps t by dt from 0 while t < T runs ⌈T/dt⌉ = ⌈T·H⌉ times: an infinite
natural, exactly determined once the residue of H is. A Hypernatural is held as a
polynomial in H with rational coefficients on each residue class of H (a Depends of
polynomials), each one integer-valued on every admitted member of its class — so ⌈H/3⌉ is
H/3, (H + 2)/3 or (H + 1)/3 as H ≡ 0, 1, 2 (mod 3), and H/2 is an infinite integer
once H is declared even.
Since H exceeds every standard natural, two values compare by the leading coefficient of their
difference, branch by branch. The residue of a value modulo k is periodic in H: an
integer-valued polynomial on an admitted class H ≡ c (mod L) is an integer combination
Σ c_j·C(t, j) of binomial coefficients in t = (H − c)/L. The term c_j·C(t, j) mod k reads
C(t, j) modulo q_j = k / gcd(c_j, k), and by Kummer’s theorem C(t, j) mod p^e has period
p^(e + ⌊log_p j⌋); so q_j times the part of lcm(1, …, j) made of primes dividing q_j is a
period, and evaluating one period of representatives gives every residue exactly. H^64 mod 2
reads two classes and C(H, 25) mod 2 the thirty-two Lucas’s theorem names.
Structs§
- Hypernatural
- An integer polynomial in the generator, by residue class. See the module documentation.
Enums§
- Floor
Error - Why the integer part of a hyperreal is not an infinite integer.