TCAP — Ξ⁺:The dynamics that reveal prime numbers

From prime-factor adjacency to a deterministic factorial prime generator

Olivier MEHAYE · TCAP · August 2026

Allegorical overview of the Ξ⁺ factorial mechanism (original French visual).

In one sentence — Ξ⁺ produces a structured value from n!; removing factors ≤ n and checking √R ≤ n yields, in favorable cases, a certified prime.

1- From Ξ to Ξ⁺: a natural extension

Ξ began with a local idea: order the prime factors of an integer and use their adjacencies. If n = p₁p₂…pₖ with p₁ ≤ p₂ ≤ … ≤ pₖ, the original transformation is Ξ(n) = p₁ + p₁p₂ + p₂p₃ + ··· + pₖ₋₁pₖ.

[embed]*Function Ξ (TCPA) : Exploring an Integer Transformation: From Prime Adjacencies to Mathematical… *Transformation by Combination of Prime Adjacencies: medium.com

Ξ⁺ is the most direct extension: it keeps the same adjacency chain while adding all prime factors themselves:

Ξ⁺(n) = (p₁ + p₂ + ··· + pₖ) + (p₁p₂ + p₂p₃ + ··· + pₖ₋₁pₖ).

Multiplicities are preserved. A prime p is a fixed point because Ξ⁺(p)=p. For composite integers, the image depends directly on the architecture of the ordered factorization.

Adjacency-based construction of Ξ⁺.

2- A dynamics strongly oriented toward primes

The transformation can be iterated: n, Ξ⁺(n), Ξ⁺²(n), … In TCAP experiments, the studied orbits very frequently reach a prime. Only one composite fixed point, 27, was observed in the large control campaigns discussed in the project, and no cycle was observed for Ξ⁺ itself.

This motivates the dynamical study, but it is not used as a proof in the factorial generator. The certificate below relies only on divisibility and the elementary square-root bound.

3- Why factorials change the problem

The factorial n! contains every prime p ≤ n, with known multiplicities. The huge integer n! does not need to be constructed: for each prime p ≤ n, Legendre’s formula gives vₚ(n!) = ⌊n/p⌋ + ⌊n/p²⌋ + ···.

Thus Ξ⁺(n!) can be computed directly from prime blocks. For a block in which p occurs e times, the contribution is ep + (e−1)p²; between consecutive primes p and q there is an additional pq term.

Example: Ξ⁺(1000!) = 60,785,481 = 3 × 23 × 880,949. After removing the factors already covered by 1000!, the remainder is 880,949, beyond the factorial reservoir.

4- The deterministic certificate

Let X = Ξ⁺(n!). Remove from X, with multiplicity, every prime factor ≤ n. Call the remaining integer R. By construction, no prime ≤ n divides R.

If R > 1 and √R ≤ n, then R is prime. Indeed, if R were composite it would have a prime divisor ≤ √R ≤ n, but every prime in that range has already been excluded. Contradiction.

R > 1 and √R ≤ n ⇒ R is prime.

“Deterministic” refers to the certificate: whenever the rule accepts a number, its primality does not depend on probability. Miller–Rabin may then be displayed as an independent a posteriori check.

Logical chain of the deterministic certificate.

5- A complete example: 9!

Ξ⁺(9!) = 145 = 5 × 29. The factor 5 belongs to 9!; after extraction, R = 29. Since √29 < 9, no possible prime divisor of 29 can remain below its square root. Therefore 29 is certified prime.

The procedure does not guess 29: it produces both the candidate and the elementary reason that certifies it.

6- Experiment through 10,000!

A campaign tested the 9,999 factorials from 2! through 10,000!. The initial gcd is 1 in 1,394 cases and non-trivial in 8,605 cases. Among the latter, 683 yield R = 1.

Among the 7,922 remainders greater than 1, 4,909 immediately satisfy √R ≤ n and are therefore certified prime: 61.97%. The other 3,013 cross the boundary of the simple certificate. An independent experimental check classifies them as 2,504 primes and 509 composites.

The essential point is that ambiguous cases are never promoted to primes: they remain outside the certified output.

Experimental distribution of remainders through 10,000!.

7- Composite cases define the boundary

For 52!, X = 13,359 = 3 × 4,453 and R = 4,453 = 61 × 73. Both factors are greater than 52, so 52! cannot eliminate them. Since R > 52², the square-root certificate does not apply.

This is not a false positive; it is simply an uncertified case. The n² bound is therefore the exact logical boundary of the elementary certificate.

8-The geometry of the generator

For every certified prime p, place the factorial rank k that produced it on the horizontal axis and p on the vertical axis. The cloud directly shows how far Ξ⁺ projects candidates.

The upper edge is exactly the certificate boundary: √p ≤ k is equivalent to p ≤ k². In the (k,p) plane, the directly certifiable region lies below the parabola p = k². This envelope is not a statistical fit; it is the geometric form of the proof.

Certified primes p as a function of factorial rank k.

The parabola p = k²: exact geometric boundary of the certificate.

9- What the generator does — and does not claim

The generator does not first select an arbitrary integer and then apply a primality test. Ξ⁺(n!) first produces a structured value; the factorial environment removes small factors; the quadratic bound then decides whether the remainder is certifiable.

It is therefore a deterministic generator with partial output: it does not guarantee a prime for every n. Frequencies observed through 10,000! are finite experimental results, not an asymptotic law.

10- Open questions

Why does Ξ⁺(n!) so often produce a prime remainder after factors ≤ n are removed? Does the cloud below p = k² contain arithmetic subfamilies or density laws specific to Ξ⁺? How far can the direct computation of Ξ⁺(n!) be pushed efficiently without ever constructing n! itself?

A further direction is dynamical: a new prime may appear immediately after extraction and another prime may later be reached by the orbit. That extension should remain separate from the elementary certificate until a general theory of Ξ⁺ attractors is established.

Conclusion

Ξ⁺ connects a dynamics on ordered factorizations, the structure of n!, and an elementary certificate obtained by excluding small factors. The mechanism is compact: compute Ξ⁺(n!), remove already-covered factors, then test √R ≤ n. Whenever this condition holds, the remainder is necessarily prime. The 4,909 certified outputs observed through 10,000! show that the phenomenon is not marginal, while its deeper distribution remains open.

Methodological note

The identities defining Ξ⁺ and the quadratic-bound certificate are mathematical arguments. Counts, percentages, orbit behavior and graphical distributions reported here come from finite TCAP experiments and are not asymptotic claims.

Minimal script — certified-prime output

The console prints only primes satisfying the √R ≤ n certificate and confirmed a posteriori by Miller–Rabin, separated by commas.

from math import isqrt

from sympy import primerange

def vfact(n, p):

e = 0

q = p

while q <= n:

e += n // q

q = p*

return e

def xi_fact(n, primes):

ps = [p for p in primes if p <= n]

x = 0

for p in ps:

e = vfact(n, p)

x += ep + (e-1)pp*

for a, b in zip(ps, ps[1:]):

x += ab*

return x

def miller_rabin(n):

if n < 2:

return False

for p in (2,3,5,7,11,13,17,19,23,29,31,37):

if n == p:

return True

if n % p == 0:

return False

d, s = n-1, 0

while d % 2 == 0:

d //= 2

s += 1

for a in (2,325,9375,28178,450775,9780504,1795265022):

if a % n == 0:

continue

x = pow(a, d, n)

if x in (1, n-1):

continue

for _ in range(s-1):

x = xx % n*

if x == n-1:

break

else:

return False

return True

N = int(input("Up to n! with n = "))

primes = list(primerange(2, N + 1))

results = []

for n in range(2, N + 1):

r = xi_fact(n, primes)

for p in primes:

if p > n:

break

while r % p == 0:

r //= p

if r > 1 and isqrt(r) <= n and miller_rabin(r):

results.append(r)

print(", ".join(map(str, results)))

O. MEHAYE